ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssxp2 Unicode version

Theorem ssxp2 4808
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 4802 . . . . . 6  |-  ( E. x  x  e.  C  ->  ran  ( C  X.  A )  =  A )
21adantr 270 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  =  A )
3 rnss 4612 . . . . . 6  |-  ( ( C  X.  A ) 
C_  ( C  X.  B )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
43adantl 271 . . . . 5  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  ran  ( C  X.  A
)  C_  ran  ( C  X.  B ) )
52, 4eqsstr3d 3043 . . . 4  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_ 
ran  ( C  X.  B ) )
6 rnxpss 4804 . . . 4  |-  ran  ( C  X.  B )  C_  B
75, 6syl6ss 3020 . . 3  |-  ( ( E. x  x  e.  C  /\  ( C  X.  A )  C_  ( C  X.  B
) )  ->  A  C_  B )
87ex 113 . 2  |-  ( E. x  x  e.  C  ->  ( ( C  X.  A )  C_  ( C  X.  B )  ->  A  C_  B ) )
9 xpss2 4497 . 2  |-  ( A 
C_  B  ->  ( C  X.  A )  C_  ( C  X.  B
) )
108, 9impbid1 140 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 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2982    X. cxp 4389   ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  xpcanm  4810
  Copyright terms: Public domain W3C validator