Theorem en3 4403

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 1475	. 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 4401	. 2 |- (A = A -> A ~~ B)
11	1, 10	ax-mp 7	1 |- A ~~ B

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619   ~~ cen 4364

This theorem is referenced by:  nn0ennn 7497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

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