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Theorem xpsndisj 5237
Description: Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
xpsndisj  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )

Proof of Theorem xpsndisj
StepHypRef Expression
1 disjsn2 3813 . 2  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2 xpdisj2 5236 . 2  |-  ( ( { B }  i^i  { D } )  =  (/)  ->  ( ( A  X.  { B }
)  i^i  ( C  X.  { D } ) )  =  (/) )
31, 2syl 16 1  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    =/= wne 2551    i^i cin 3263   (/)c0 3572   {csn 3758    X. cxp 4817
This theorem is referenced by:  xp01disj  6677  unxpdom2  7254  sucxpdom  7255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827
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