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Theorem xpsndisj 4773
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 3457 . 2  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2 xpdisj2 4772 . 2  |-  ( ( { B }  i^i  { D } )  =  (/)  ->  ( ( A  X.  { B }
)  i^i  ( C  X.  { D } ) )  =  (/) )
31, 2syl 14 1  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    =/= wne 2246    i^i cin 2973   (/)c0 3252   {csn 3400    X. cxp 4363
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-in1 577  ax-in2 578  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 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373
This theorem is referenced by:  xp01disj  6075
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