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Theorem ssxp1 4757
Description: Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
Assertion
Ref Expression
ssxp1  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  C_  ( B  X.  C )  <->  A  C_  B
) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssxp1
StepHypRef Expression
1 dmxpm 4555 . . . . . 6  |-  ( E. x  x  e.  C  ->  dom  ( A  X.  C )  =  A )
21adantr 261 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( A  X.  C )  C_  ( B  X.  C
) )  ->  dom  ( A  X.  C
)  =  A )
3 dmss 4534 . . . . . 6  |-  ( ( A  X.  C ) 
C_  ( B  X.  C )  ->  dom  ( A  X.  C
)  C_  dom  ( B  X.  C ) )
43adantl 262 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( A  X.  C )  C_  ( B  X.  C
) )  ->  dom  ( A  X.  C
)  C_  dom  ( B  X.  C ) )
52, 4eqsstr3d 2980 . . . 4  |-  ( ( E. x  x  e.  C  /\  ( A  X.  C )  C_  ( B  X.  C
) )  ->  A  C_ 
dom  ( B  X.  C ) )
6 dmxpss 4753 . . . 4  |-  dom  ( B  X.  C )  C_  B
75, 6syl6ss 2957 . . 3  |-  ( ( E. x  x  e.  C  /\  ( A  X.  C )  C_  ( B  X.  C
) )  ->  A  C_  B )
87ex 108 . 2  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  C_  ( B  X.  C )  ->  A  C_  B ) )
9 xpss1 4448 . 2  |-  ( A 
C_  B  ->  ( A  X.  C )  C_  ( B  X.  C
) )
108, 9impbid1 130 1  |-  ( E. x  x  e.  C  ->  ( ( A  X.  C )  C_  ( B  X.  C )  <->  A  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393    C_ wss 2917    X. cxp 4343   dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-dm 4355
This theorem is referenced by:  xpcan2m  4761
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