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Theorem xpcan2m 4986
Description: Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
xpcan2m  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  =  ( B  X.  C )  <-> 
A  =  B ) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem xpcan2m
StepHypRef Expression
1 ssxp1 4982 . . 3  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  C_  ( B  X.  C )  <->  A  C_  B
) )
2 ssxp1 4982 . . 3  |-  ( E. x  x  e.  C  ->  ( ( B  X.  C )  C_  ( A  X.  C )  <->  B  C_  A
) )
31, 2anbi12d 465 . 2  |-  ( E. x  x  e.  C  ->  ( ( ( A  X.  C )  C_  ( B  X.  C
)  /\  ( B  X.  C )  C_  ( A  X.  C ) )  <-> 
( A  C_  B  /\  B  C_  A ) ) )
4 eqss 3116 . 2  |-  ( ( A  X.  C )  =  ( B  X.  C )  <->  ( ( A  X.  C )  C_  ( B  X.  C
)  /\  ( B  X.  C )  C_  ( A  X.  C ) ) )
5 eqss 3116 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
63, 4, 53bitr4g 222 1  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  =  ( B  X.  C )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481    C_ wss 3075    X. cxp 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-dm 4556
This theorem is referenced by: (None)
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