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Theorem xpsndisj 5105
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 3696 . 2  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2 xpdisj2 5104 . 2  |-  ( ( { B }  i^i  { D } )  =  (/)  ->  ( ( A  X.  { B }
)  i^i  ( C  X.  { D } ) )  =  (/) )
31, 2syl 15 1  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    =/= wne 2448    i^i cin 3153   (/)c0 3457   {csn 3642    X. cxp 4689
This theorem is referenced by:  xp01disj  6497  unxpdom2  7073  sucxpdom  7074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-cnv 4699
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