Theorem xpcanm 4780

Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)

Assertion

Ref	Expression
xpcanm	|- ( E. x x e. C -> ( ( C X. A ) = ( C X. B ) <-> A = B ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem xpcanm

Step	Hyp	Ref	Expression
1		ssxp2 4778	. . 3 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )
2		ssxp2 4778	. . 3 |- ( E. x x e. C -> ( ( C X. B ) C_ ( C X. A ) <-> B C_ A ) )
3	1, 2	anbi12d 456	. 2 |- ( E. x x e. C -> ( ( ( C X. A ) C_ ( C X. B ) /\ ( C X. B ) C_ ( C X. A ) ) <-> ( A C_ B /\ B C_ A ) ) )
4		eqss 3014	. 2 |- ( ( C X. A ) = ( C X. B ) <-> ( ( C X. A ) C_ ( C X. B ) /\ ( C X. B ) C_ ( C X. A ) ) )
5		eqss 3014	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
6	3, 4, 5	3bitr4g 221	1 |- ( E. x x e. C -> ( ( C X. A ) = ( C X. B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   C_ wss 2973   X. cxp 4361

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-rex 2354  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-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by: (None)