Theorem en3 3306

Description: Equinumerosity inference from an implicit one-to-one onto function.

Hypotheses

Ref	Expression
en3.1	|- A e. V
en3.2	|- (x e. A -> C e. B)
en3.3	|- (y e. B -> D e. A)
en3.4	|- ((x e. A /\ y e. B) -> (x = D <-> y = C))

Assertion

Ref	Expression
en3	|- A ~~ B

Distinct variable group(s):   x,y,A   x,B,y   y,C   x,D

Proof of Theorem en3

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 |- A = A
2		en3.1	. . . 4 |- A e. V
3	2	a1i 7	. . 3 |- (A = A -> A e. V)
4		en3.2	. . . 4 |- (x e. A -> C e. B)
5	4	a1i 7	. . 3 |- (A = A -> (x e. A -> C e. B))
6		en3.3	. . . 4 |- (y e. B -> D e. A)
7	6	a1i 7	. . 3 |- (A = A -> (y e. B -> D e. A))
8		en3.4	. . . 4 |- ((x e. A /\ y e. B) -> (x = D <-> y = C))
9	8	a1i 7	. . 3 |- (A = A -> ((x e. A /\ y e. B) -> (x = D <-> y = C)))
10	3, 5, 7, 9	en3d 3304	. 2 |- (A = A -> A ~~ B)
11	1, 10	ax-mp 6	1 |- A ~~ B

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054   ~~ cen 3271

This theorem is referenced by:  nn0ennn 4925

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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