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Theorem ssxp2 4881
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp2  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  <->  A  C_  B
) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssxp2
StepHypRef Expression
1 rnxpm 4873 . . . . . 6  |-  ( E. x  x  e.  C  ->  ran  ( C  X.  A )  =  A )
21adantr 271 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  =  A )
3 rnss 4678 . . . . . 6  |-  ( ( C  X.  A ) 
C_  ( C  X.  B )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
43adantl 272 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
52, 4eqsstr3d 3062 . . . 4  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_ 
ran  ( C  X.  B ) )
6 rnxpss 4875 . . . 4  |-  ran  ( C  X.  B )  C_  B
75, 6syl6ss 3038 . . 3  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_  B )
87ex 114 . 2  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  ->  A  C_  B ) )
9 xpss2 4562 . 2  |-  ( A 
C_  B  ->  ( C  X.  A )  C_  ( C  X.  B
) )
108, 9impbid1 141 1  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  <->  A  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290   E.wex 1427    e. wcel 1439    C_ wss 3000    X. cxp 4449   ran crn 4452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-rel 4458  df-cnv 4459  df-dm 4461  df-rn 4462
This theorem is referenced by:  xpcanm  4883
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