--- a/ordinal-definable.agda	Mon Jun 10 09:50:52 2019 +0900
+++ b/ordinal-definable.agda	Wed Jun 12 10:45:00 2019 +0900
@@ -45,28 +45,27 @@
 od∅ : {n : Level} → OD {n} 
 od∅ {n} = record { def = λ _ → Lift n ⊥ }
 
-
 postulate      
   -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
   od→ord : {n : Level} → OD {n} → Ordinal {n}
   ord→od : {n : Level} → Ordinal {n} → OD {n} 
-  c<→o<  : {n : Level} {x y : OD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
-  o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
   oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
   diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
   -- supermum as Replacement Axiom
   sup-o  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
   sup-o< : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → ∀ {x : Ordinal {n}} →  ψ x  o<  sup-o ψ 
-  -- a property of supermum required in Power Set Axiom
+  -- a contra-position of minimality of supermum 
   sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
   sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
-  -- sup-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( sup-o ψ ) 
 
 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
 _∋_ {n} a x  = def a ( od→ord x )
 
 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n
-x c< a = a ∋ x 
+x c< a =  od→ord x o< od→ord a
+
+postulate      
+   o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y  → ord→od x c< ord→od y
 
 _c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
 a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
@@ -74,15 +73,17 @@
 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z  →  X ≡ x  →  def z x
 def-subst df refl refl = df
 
+-- sup-min : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → {z : Ordinal {n}}  →  ψ z  o<  z  →   sup-o ψ  o< osuc z
+
 sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
 
 sup-c< : {n : Level } → ( ψ : OD {n} →  OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )}
-        ( o<→c< sup-o< ) refl (cong ( λ k → od→ord (ψ k) ) oiso)
+        {!!} refl (cong ( λ k → od→ord (ψ k) ) oiso)
 
 ∅1 : {n : Level} →  ( x : OD {n} )  → ¬ ( x c< od∅ {n} )
-∅1 {n} x (lift ())
+∅1 {n} x = {!!}
 
 ∅3 : {n : Level} →  { x : Ordinal {n}}  → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
@@ -102,18 +103,13 @@
    c3 lx (OSuc .lx x₁) d not | t | ()
 
 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z  ∋ x
-transitive  {n} {z} {y} {x} z∋y x∋y  with  ordtrans ( c<→o< {suc n} {x} {y} x∋y ) (  c<→o< {suc n} {y} {z} z∋y ) 
+transitive  {n} {z} {y} {x} z∋y x∋y  with  ordtrans {!!} {!!} 
 ... | t = lemma0 (lemma t) where
    lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x)
-   lemma xo<z = o<→c< xo<z
+   lemma xo<z = {!!}
    lemma0 :  def ( ord→od ( od→ord z )) ( od→ord x) →  def z (od→ord x)
    lemma0 dz  = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso)  refl
 
-record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
-  field
-     mino : Ordinal {n}
-     min<x :  mino o< x
-
 ∅5 : {n : Level} →  { x : Ordinal {n} }  → ¬ ( x  ≡ o∅ {n} ) → o∅ {n} o< x
 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) 
 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
@@ -153,12 +149,12 @@
 
 o<→o> : {n : Level} →  { x y : OD {n} } →  (x == y) → (od→ord x ) o< ( od→ord y) → ⊥
 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with
-     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl )
-... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl )
+... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!}
 ... | ()
 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with
-     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl )
-... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl )
+... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!}
 ... | ()
 
 ==→o≡ : {n : Level} →  { x y : Ordinal {suc n} } → ord→od x == ord→od y →  x ≡ y 
@@ -172,8 +168,8 @@
     lemma :  ord→od x == record { def = λ z → z o< x }
     eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where 
         t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
-        t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso))
-    eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl
+        t = {!!}
+    eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} {!!} refl refl
 
 od≡-def : {n : Level} →  { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } 
 od≡-def {n} {x} = subst (λ k  → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} ))
@@ -190,19 +186,19 @@
 ∋→o< : {n : Level} →  { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a
 ∋→o< {n} {a} {x} lt = t where
          t : (od→ord x) o< (od→ord a)
-         t = c<→o< {suc n} {x} {a} lt 
+         t = {!!}
 
 o<∋→ : {n : Level} →  { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x 
 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t  where
          t : def (ord→od (od→ord a)) (od→ord x)
-         t = o<→c< {suc n} {od→ord x} {od→ord a} lt 
+         t = {!!}
 
 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
     lemma :  o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
-    lemma lt with def-subst (o<→c< lt) oiso refl
-    lemma lt | lift ()
+    lemma lt with def-subst {!!} oiso refl
+    lemma lt | t = {!!}
 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
 
@@ -210,21 +206,21 @@
 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
 
 o<→¬c> : {n : Level} →  { x y : OD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (y c< x )
-o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
+o<→¬c> {n} {x} {y} olt clt = o<> olt {!!} where
 
 o≡→¬c< : {n : Level} →  { x y : OD {n} } →  (od→ord x ) ≡ ( od→ord y) →   ¬ x c< y
-o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) 
+o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) lt  
 
 tri-c< : {n : Level} →  Trichotomous _==_ (_c<_ {suc n})
 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) 
-tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a )
+tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst {!!} oiso refl) (o<→¬== a) ( o<→¬c> a )
 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b))
-tri-c< {n} x y | tri> ¬a ¬b c = tri>  ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso refl)
+tri-c< {n} x y | tri> ¬a ¬b c = tri>  ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst {!!} oiso refl)
 
 c<> : {n : Level } { x y : OD {suc n}} → x c<  y  → y c< x  →  ⊥
 c<> {n} {x} {y} x<y y<x with tri-c< x y
 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
-c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y )
+c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b x<y 
 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
 
 ∅< : {n : Level} →  { x y : OD {n} } → def x (od→ord y ) → ¬ (  x  == od∅ {n} )
@@ -232,16 +228,16 @@
 ∅< {n} {x} {y} d eq | lift ()
        
 ∅6 : {n : Level} →  { x : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
-∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
+∅6 {n} {x} x∋x = c<> {n} {x} {x} {!!} {!!}
 
 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y  → (def A y → def B y)  → def A x → def B x
 def-iso refl t = t
 
 is-∋ : {n : Level} →  ( x y : OD {suc n} ) → Dec ( x ∋ y )
 is-∋ {n} x y with tri-c< x y
-is-∋ {n} x y | tri< a ¬b ¬c = no ¬c
-is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c
-is-∋ {n} x y | tri> ¬a ¬b c = yes c
+is-∋ {n} x y | tri< a ¬b ¬c = no {!!}
+is-∋ {n} x y | tri≈ ¬a b ¬c = no {!!}
+is-∋ {n} x y | tri> ¬a ¬b c = yes {!!}
 
 is-o∅ : {n : Level} →  ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
@@ -258,7 +254,7 @@
          lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅
          lemma1 = cong ( λ k → od→ord k ) o∅≡od∅
      lemma o∅ ne | yes refl | ()
-     lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) )  
+     lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ {!!}
 
 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
 -- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
@@ -311,10 +307,12 @@
     A ∈ B = B ∋ A
     _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set (suc n)
     _⊆_ A B {x} = A ∋ x →  B ∋ x
-    -- _∩_ : ( A B : ZFSet  ) → ZFSet
-    -- A ∩ B = Select (A , B) (  λ x → ( A ∋ x ) ∧ (B ∋ x) )
+    _∩_ : ( A B : ZFSet  ) → ZFSet
+    A ∩ B = Select (A , B) (  λ x → ( A ∋ x ) ∧ (B ∋ x) )
     -- _∪_ : ( A B : ZFSet  ) → ZFSet
     -- A ∪ B = Select (A , B) (  λ x → (A ∋ x)  ∨ ( B ∋ x ) )
+    ｛_｝ : ZFSet → ZFSet
+    ｛ x ｝ = ( x ,  x )
     infixr  200 _∈_
     -- infixr  230 _∩_ _∪_
     infixr  220 _⊆_
@@ -337,7 +335,6 @@
        ;   replacement = replacement
      } where
          open _∧_ 
-         open Minimumo
          pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧  ((A , B) ∋ B)
          proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
          proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
@@ -347,7 +344,7 @@
          --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
          --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )       Power X is a sup of all subset of A
          --
-         --  if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t 
+         --  if Power A ∋ t, from a minimulity of sup, there is osuc ZFSubset A ∋ t 
          --    then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
          --    In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity
          --
@@ -356,19 +353,16 @@
               minsup :  OD
               minsup =  ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) 
               lemma-t : csuc minsup ∋ t
-              lemma-t = o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) 
+              lemma-t = {!!}
               lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))  ∋ x
-              lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso  )
-              lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl
-              lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) t∋x
+              lemma-s = {!!}
          -- 
          -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
          -- Power A is a sup of ZFSubset A t, so Power A ∋ t
          -- 
          power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
          power← A t t→A  = def-subst {suc n} {_} {_} {Power A} {od→ord t}
-                  ( o<→c< {suc n} {od→ord (ZFSubset A (ord→od (od→ord t)) )} {sup-o (λ x → od→ord (ZFSubset A (ord→od x)))}
-                      lemma ) refl lemma1 where
+                  {!!} refl lemma1 where
               lemma-eq :  ZFSubset A t == t
               eq→ lemma-eq {z} w = proj2 w 
               eq← lemma-eq {z} w = record { proj2 = w  ;
@@ -380,17 +374,17 @@
          union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z
          union-lemma-u {X} {z} U>z = lemma <-osuc where
              lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
-             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} (o<→c< lt) refl refl
+             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} {!!} refl refl
          union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
          union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
-         union→ X y u xx | tri< a ¬b ¬c with  osuc-< a (c<→o< (proj2 xx))
+         union→ X y u xx | tri< a ¬b ¬c with  osuc-< a {!!}
          union→ X y u xx | tri< a ¬b ¬c | ()
-         union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where
+         union→ X y u xx | tri≈ ¬a b ¬c = lemma b {!!} where
              lemma : {oX ou ooy : Ordinal {suc n}} →  ou ≡ ooy  → ou o< oX   → ooy  o< oX
              lemma refl lt = lt
-         union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) 
+         union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c {!!} 
          union← :  (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z )
-         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
+         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
          ψiso :  {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y   → ψ y
          ψiso {ψ} t refl = t
          selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
@@ -419,7 +413,7 @@
          xxx-union : {x  : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
          xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where
              lemma1 : {x  : OD {suc n}} → od→ord x o< od→ord (x , x)
-             lemma1 {x} = c<→o< ( proj1 (pair x x ) )
+             lemma1 {x} = {!!}
              lemma2 : {x  : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
              lemma2 = trans ( cong ( λ k →  od→ord k ) xx-union ) (sym ≡-def)
              lemma : {x  : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
@@ -438,7 +432,7 @@
          infinite = ord→od ( omega )
          infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
          infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅}
-              (o<→c< ( case1 (s≤s z≤n )))  refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k →  od→ord k) o∅≡od∅ ))
+              {!!}  refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k →  od→ord k) o∅≡od∅ ))
          infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega
          infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where
               t  : od→ord x o< od→ord (ord→od (omega))
@@ -456,4 +450,15 @@
               lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
               lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
               lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
+         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set
+         -- ∀ z [ ∀ x ( x ∈ z  → ¬ ( x ≈ ∅ ) )  ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y )  → x ∩ y ≈ ∅  ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ]
+         record Choice (z : OD {suc n}) : Set (suc (suc n)) where
+             field
+                 u : {x : OD {suc n}} ( x∈z  : x ∈ z ) → OD {suc n}
+                 t : {x : OD {suc n}} ( x∈z  : x ∈ z ) → (x : OD {suc n} ) → OD {suc n}
+                 choice : { x : OD {suc n} } → ( x∈z  : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝
+         -- choice : {x :  OD {suc n}} ( x ∈ z  → ¬ ( x ≈ ∅ ) ) →
+         -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) →  choice ¬x∅ A∈X ∈ A 
+         -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}
 
+