Theorem xpid11m 4557

Description: The cross product of a class with itself is one-to-one. (Contributed by Jim Kingdon, 8-Dec-2018.)

Assertion

Ref	Expression
xpid11m	|- ( ( E. x x e. A /\ E. x x e. B ) -> ( ( A X. A ) = ( B X. B ) <-> A = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem xpid11m

Step	Hyp	Ref	Expression
1		dmxpm 4555	. . . . . 6 |- ( E. x x e. A -> dom ( A X. A ) = A )
2	1	adantr 261	. . . . 5 |- ( ( E. x x e. A /\ E. x x e. B ) -> dom ( A X. A ) = A )
3		dmeq 4535	. . . . 5 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
4	2, 3	sylan9req 2093	. . . 4 |- ( ( ( E. x x e. A /\ E. x x e. B ) /\ ( A X. A ) = ( B X. B ) ) -> A = dom ( B X. B ) )
5		dmxpm 4555	. . . . 5 |- ( E. x x e. B -> dom ( B X. B ) = B )
6	5	ad2antlr 458	. . . 4 |- ( ( ( E. x x e. A /\ E. x x e. B ) /\ ( A X. A ) = ( B X. B ) ) -> dom ( B X. B ) = B )
7	4, 6	eqtrd 2072	. . 3 |- ( ( ( E. x x e. A /\ E. x x e. B ) /\ ( A X. A ) = ( B X. B ) ) -> A = B )
8	7	ex 108	. 2 |- ( ( E. x x e. A /\ E. x x e. B ) -> ( ( A X. A ) = ( B X. B ) -> A = B ) )
9		xpeq12 4364	. . 3 |- ( ( A = B /\ A = B ) -> ( A X. A ) = ( B X. B ) )
10	9	anidms 377	. 2 |- ( A = B -> ( A X. A ) = ( B X. B ) )
11	8, 10	impbid1 130	1 |- ( ( E. x x e. A /\ E. x x e. B ) -> ( ( A X. A ) = ( B X. B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   X. cxp 4343   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355

This theorem is referenced by: (None)