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Theorem difxp2 5362
Description: Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
difxp2  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )

Proof of Theorem difxp2
StepHypRef Expression
1 difxp 5360 . 2  |-  ( ( A  X.  B ) 
\  ( A  X.  C ) )  =  ( ( ( A 
\  A )  X.  B )  u.  ( A  X.  ( B  \  C ) ) )
2 difid 3845 . . . . 5  |-  ( A 
\  A )  =  (/)
32xpeq1i 4958 . . . 4  |-  ( ( A  \  A )  X.  B )  =  ( (/)  X.  B
)
4 0xp 5015 . . . 4  |-  ( (/)  X.  B )  =  (/)
53, 4eqtri 2480 . . 3  |-  ( ( A  \  A )  X.  B )  =  (/)
65uneq1i 3604 . 2  |-  ( ( ( A  \  A
)  X.  B )  u.  ( A  X.  ( B  \  C ) ) )  =  (
(/)  u.  ( A  X.  ( B  \  C
) ) )
7 uncom 3598 . . 3  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( ( A  X.  ( B 
\  C ) )  u.  (/) )
8 un0 3760 . . 3  |-  ( ( A  X.  ( B 
\  C ) )  u.  (/) )  =  ( A  X.  ( B 
\  C ) )
97, 8eqtri 2480 . 2  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( A  X.  ( B  \  C ) )
101, 6, 93eqtrri 2485 1  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    \ cdif 3423    u. cun 3424   (/)c0 3735    X. cxp 4936
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-opab 4449  df-xp 4944  df-rel 4945
This theorem is referenced by:  imadifxp  26073  sxbrsigalem2  26835
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