Theorem en3i 6737

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)

Hypotheses

Ref	Expression
en3i.1	|- A e. _V
en3i.2	|- B e. _V
en3i.3	|- ( x e. A -> C e. B )
en3i.4	|- ( y e. B -> D e. A )
en3i.5	|- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) )

Assertion

Ref	Expression
en3i	|- A ~~ B

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D

Allowed substitution hints:   C( x)   D( y)

Proof of Theorem en3i

Step	Hyp	Ref	Expression
1		en3i.1	. . . 4 |- A e. _V
2	1	a1i 9	. . 3 |- ( T. -> A e. _V )
3		en3i.2	. . . 4 |- B e. _V
4	3	a1i 9	. . 3 |- ( T. -> B e. _V )
5		en3i.3	. . . 4 |- ( x e. A -> C e. B )
6	5	a1i 9	. . 3 |- ( T. -> ( x e. A -> C e. B ) )
7		en3i.4	. . . 4 |- ( y e. B -> D e. A )
8	7	a1i 9	. . 3 |- ( T. -> ( y e. B -> D e. A ) )
9		en3i.5	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) )
10	9	a1i 9	. . 3 |- ( T. -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )
11	2, 4, 6, 8, 10	en3d 6735	. 2 |- ( T. -> A ~~ B )
12	11	mptru 1352	1 |- A ~~ B

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   T. wtru 1344   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707

This theorem is referenced by:  xpmapenlem  6815  nn0ennn  10368  oddennn  12325  evenennn  12326  znnen  12331