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Theorem ssxp2 5078
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 5070 . . . . . 6  |-  ( E. x  x  e.  C  ->  ran  ( C  X.  A )  =  A )
21adantr 276 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  =  A )
3 rnss 4869 . . . . . 6  |-  ( ( C  X.  A ) 
C_  ( C  X.  B )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
43adantl 277 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
52, 4eqsstrrd 3204 . . . 4  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_ 
ran  ( C  X.  B ) )
6 rnxpss 5072 . . . 4  |-  ran  ( C  X.  B )  C_  B
75, 6sstrdi 3179 . . 3  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_  B )
87ex 115 . 2  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  ->  A  C_  B ) )
9 xpss2 4749 . 2  |-  ( A 
C_  B  ->  ( C  X.  A )  C_  ( C  X.  B
) )
108, 9impbid1 142 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 104    <-> wb 105    = wceq 1363   E.wex 1502    e. wcel 2158    C_ wss 3141    X. cxp 4636   ran crn 4639
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  xpcanm  5080
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