Theorem xpcanm 4870

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 4868	. . 3 |- ( E. x x e. C -> ( ( C X. A ) C_ ( C X. B ) <-> A C_ B ) )
2		ssxp2 4868	. . 3 |- ( E. x x e. C -> ( ( C X. B ) C_ ( C X. A ) <-> B C_ A ) )
3	1, 2	anbi12d 457	. 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 3040	. 2 |- ( ( C X. A ) = ( C X. B ) <-> ( ( C X. A ) C_ ( C X. B ) /\ ( C X. B ) C_ ( C X. A ) ) )
5		eqss 3040	. 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 1289   E.wex 1426   e. wcel 1438   C_ wss 2999   X. cxp 4436

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449

This theorem is referenced by: (None)