Theorem en3 4393

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 groups:   x,y,A   x,B,y   y,C   x,D

Proof of Theorem en3

Step	Hyp	Ref	Expression
1		eqid 1474	. 2 |- A = A
2		en3.1	. . . 4 |- A e. V
3	2	a1i 8	. . 3 |- (A = A -> A e. V)
4		en3.2	. . . 4 |- (x e. A -> C e. B)
5	4	a1i 8	. . 3 |- (A = A -> (x e. A -> C e. B))
6		en3.3	. . . 4 |- (y e. B -> D e. A)
7	6	a1i 8	. . 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 8	. . 3 |- (A = A -> ((x e. A /\ y e. B) -> (x = D <-> y = C)))
10	3, 5, 7, 9	en3d 4391	. 2 |- (A = A -> A ~~ B)
11	1, 10	ax-mp 7	1 |- A ~~ B

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808   class class class wbr 2615   ~~ cen 4357

This theorem is referenced by:  nn0ennn 7456

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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