Theorem xpid11m 4575

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 4573	. . . . . 6 |- ( E. x x e. A -> dom ( A X. A ) = A )
2	1	adantr 270	. . . . 5 |- ( ( E. x x e. A /\ E. x x e. B ) -> dom ( A X. A ) = A )
3		dmeq 4553	. . . . 5 |- ( ( A X. A ) = ( B X. B ) -> dom ( A X. A ) = dom ( B X. B ) )
4	2, 3	sylan9req 2134	. . . 4 |- ( ( ( E. x x e. A /\ E. x x e. B ) /\ ( A X. A ) = ( B X. B ) ) -> A = dom ( B X. B ) )
5		dmxpm 4573	. . . . 5 |- ( E. x x e. B -> dom ( B X. B ) = B )
6	5	ad2antlr 472	. . . 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 2113	. . 3 |- ( ( ( E. x x e. A /\ E. x x e. B ) /\ ( A X. A ) = ( B X. B ) ) -> A = B )
8	7	ex 113	. 2 |- ( ( E. x x e. A /\ E. x x e. B ) -> ( ( A X. A ) = ( B X. B ) -> A = B ) )
9		xpeq12 4382	. . 3 |- ( ( A = B /\ A = B ) -> ( A X. A ) = ( B X. B ) )
10	9	anidms 389	. 2 |- ( A = B -> ( A X. A ) = ( B X. B ) )
11	8, 10	impbid1 140	1 |- ( ( E. x x e. A /\ E. x x e. B ) -> ( ( A X. A ) = ( B X. B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   X. cxp 4361   dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-dm 4373

This theorem is referenced by: (None)