Theorem xp11 5569

Description: The Cartesian product of nonempty classes is one-to-one. (Contributed by NM, 31-May-2008.)

Assertion

Ref	Expression
xp11	|- ( ( A =/= (/) /\ B =/= (/) ) -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) )

Proof of Theorem xp11

Step	Hyp	Ref	Expression
1		xpnz 5553	. . 3 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )
2		anidm 676	. . . . . 6 |- ( ( ( A X. B ) =/= (/) /\ ( A X. B ) =/= (/) ) <-> ( A X. B ) =/= (/) )
3		neeq1 2856	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( ( A X. B ) =/= (/) <-> ( C X. D ) =/= (/) ) )
4	3	anbi2d 740	. . . . . 6 |- ( ( A X. B ) = ( C X. D ) -> ( ( ( A X. B ) =/= (/) /\ ( A X. B ) =/= (/) ) <-> ( ( A X. B ) =/= (/) /\ ( C X. D ) =/= (/) ) ) )
5	2, 4	syl5bbr 274	. . . . 5 |- ( ( A X. B ) = ( C X. D ) -> ( ( A X. B ) =/= (/) <-> ( ( A X. B ) =/= (/) /\ ( C X. D ) =/= (/) ) ) )
6		eqimss 3657	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( A X. B ) C_ ( C X. D ) )
7		ssxpb 5568	. . . . . . . 8 |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) <-> ( A C_ C /\ B C_ D ) ) )
8	6, 7	syl5ibcom 235	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( ( A X. B ) =/= (/) -> ( A C_ C /\ B C_ D ) ) )
9		eqimss2 3658	. . . . . . . 8 |- ( ( A X. B ) = ( C X. D ) -> ( C X. D ) C_ ( A X. B ) )
10		ssxpb 5568	. . . . . . . 8 |- ( ( C X. D ) =/= (/) -> ( ( C X. D ) C_ ( A X. B ) <-> ( C C_ A /\ D C_ B ) ) )
11	9, 10	syl5ibcom 235	. . . . . . 7 |- ( ( A X. B ) = ( C X. D ) -> ( ( C X. D ) =/= (/) -> ( C C_ A /\ D C_ B ) ) )
12	8, 11	anim12d 586	. . . . . 6 |- ( ( A X. B ) = ( C X. D ) -> ( ( ( A X. B ) =/= (/) /\ ( C X. D ) =/= (/) ) -> ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) ) )
13		an4 865	. . . . . . 7 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
14		eqss 3618	. . . . . . . 8 |- ( A = C <-> ( A C_ C /\ C C_ A ) )
15		eqss 3618	. . . . . . . 8 |- ( B = D <-> ( B C_ D /\ D C_ B ) )
16	14, 15	anbi12i 733	. . . . . . 7 |- ( ( A = C /\ B = D ) <-> ( ( A C_ C /\ C C_ A ) /\ ( B C_ D /\ D C_ B ) ) )
17	13, 16	bitr4i 267	. . . . . 6 |- ( ( ( A C_ C /\ B C_ D ) /\ ( C C_ A /\ D C_ B ) ) <-> ( A = C /\ B = D ) )
18	12, 17	syl6ib 241	. . . . 5 |- ( ( A X. B ) = ( C X. D ) -> ( ( ( A X. B ) =/= (/) /\ ( C X. D ) =/= (/) ) -> ( A = C /\ B = D ) ) )
19	5, 18	sylbid 230	. . . 4 |- ( ( A X. B ) = ( C X. D ) -> ( ( A X. B ) =/= (/) -> ( A = C /\ B = D ) ) )
20	19	com12 32	. . 3 |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
21	1, 20	sylbi 207	. 2 |- ( ( A =/= (/) /\ B =/= (/) ) -> ( ( A X. B ) = ( C X. D ) -> ( A = C /\ B = D ) ) )
22		xpeq12 5134	. 2 |- ( ( A = C /\ B = D ) -> ( A X. B ) = ( C X. D ) )
23	21, 22	impbid1 215	1 |- ( ( A =/= (/) /\ B =/= (/) ) -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   =/= wne 2794   C_ wss 3574   (/)c0 3915   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  xpcan  5570  xpcan2  5571  fseqdom  8849  axcc2lem  9258  lmodfopnelem1  18899