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Theorem xpcan2m 5087
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 5083 . . 3  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  C_  ( B  X.  C )  <->  A  C_  B
) )
2 ssxp1 5083 . . 3  |-  ( E. x  x  e.  C  ->  ( ( B  X.  C )  C_  ( A  X.  C )  <->  B  C_  A
) )
31, 2anbi12d 473 . 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 3185 . 2  |-  ( ( A  X.  C )  =  ( B  X.  C )  <->  ( ( A  X.  C )  C_  ( B  X.  C
)  /\  ( B  X.  C )  C_  ( A  X.  C ) ) )
5 eqss 3185 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
63, 4, 53bitr4g 223 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 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160    C_ wss 3144    X. cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-dm 4654
This theorem is referenced by: (None)
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