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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
StepHypRef 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
) )
31, 2anbi12d 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 ) )
63, 4, 53bitr4g 221 1  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  =  ( C  X.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
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)
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