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Theorem xpcanm 5048
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 5046 . . 3  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  <->  A  C_  B
) )
2 ssxp2 5046 . . 3  |-  ( E. x  x  e.  C  ->  ( ( C  X.  B )  C_  ( C  X.  A )  <->  B  C_  A
) )
31, 2anbi12d 470 . 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 3162 . 2  |-  ( ( C  X.  A )  =  ( C  X.  B )  <->  ( ( C  X.  A )  C_  ( C  X.  B
)  /\  ( C  X.  B )  C_  ( C  X.  A ) ) )
5 eqss 3162 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
63, 4, 53bitr4g 222 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 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141    C_ wss 3121    X. cxp 4607
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by: (None)
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