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Theorem difxp1 5366
Description: Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
difxp1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )

Proof of Theorem difxp1
StepHypRef Expression
1 difxp 5365 . 2  |-  ( ( A  X.  C ) 
\  ( B  X.  C ) )  =  ( ( ( A 
\  B )  X.  C )  u.  ( A  X.  ( C  \  C ) ) )
2 difid 3850 . . . . 5  |-  ( C 
\  C )  =  (/)
32xpeq2i 4964 . . . 4  |-  ( A  X.  ( C  \  C ) )  =  ( A  X.  (/) )
4 xp0 5359 . . . 4  |-  ( A  X.  (/) )  =  (/)
53, 4eqtri 2481 . . 3  |-  ( A  X.  ( C  \  C ) )  =  (/)
65uneq2i 3610 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  ( A  X.  ( C  \  C ) ) )  =  ( ( ( A  \  B )  X.  C
)  u.  (/) )
7 un0 3765 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  (/) )  =  ( ( A  \  B
)  X.  C )
81, 6, 73eqtrri 2486 1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    \ cdif 3428    u. cun 3429   (/)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-eu 2265  df-mo 2266  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-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-cnv 4951
This theorem is referenced by:  dfsup3OLD  7800  sxbrsigalem2  26840
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