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Theorem xpdisj1 5362
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
xpdisj1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )

Proof of Theorem xpdisj1
StepHypRef Expression
1 inxp 5075 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  D ) )
2 xpeq1 4957 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (
(/)  X.  ( C  i^i  D ) ) )
3 0xp 5020 . . 3  |-  ( (/)  X.  ( C  i^i  D
) )  =  (/)
42, 3syl6eq 2509 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (/) )
51, 4syl5eq 2505 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    i^i cin 3430   (/)c0 3740    X. cxp 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4454  df-xp 4949  df-rel 4950
This theorem is referenced by:  djudisj  5368  xpdisjres  26084  bj-2upln1upl  32830
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