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Theorem xpdisj1 5413
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 5124 . 2  |-  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  ( ( A  i^i  B )  X.  ( C  i^i  D ) )
2 xpeq1 5002 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (
(/)  X.  ( C  i^i  D ) ) )
3 0xp 5069 . . 3  |-  ( (/)  X.  ( C  i^i  D
) )  =  (/)
42, 3syl6eq 2511 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  B )  X.  ( C  i^i  D ) )  =  (/) )
51, 4syl5eq 2507 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  X.  C )  i^i  ( B  X.  D ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    i^i cin 3460   (/)c0 3783    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995
This theorem is referenced by:  djudisj  5419  xpdisjres  27672  esum2dlem  28324  bj-2upln1upl  35002
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