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Theorem xpsndisj 5288
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 3861 . 2  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2 xpdisj2 5287 . 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 1652    =/= wne 2598    i^i cin 3311   (/)c0 3620   {csn 3806    X. cxp 4868
This theorem is referenced by:  xp01disj  6732  unxpdom2  7309  sucxpdom  7310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878
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