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Theorem difxp2 5423
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 5421 . 2  |-  ( ( A  X.  B ) 
\  ( A  X.  C ) )  =  ( ( ( A 
\  A )  X.  B )  u.  ( A  X.  ( B  \  C ) ) )
2 difid 3882 . . . . 5  |-  ( A 
\  A )  =  (/)
32xpeq1i 5009 . . . 4  |-  ( ( A  \  A )  X.  B )  =  ( (/)  X.  B
)
4 0xp 5070 . . . 4  |-  ( (/)  X.  B )  =  (/)
53, 4eqtri 2472 . . 3  |-  ( ( A  \  A )  X.  B )  =  (/)
65uneq1i 3639 . 2  |-  ( ( ( A  \  A
)  X.  B )  u.  ( A  X.  ( B  \  C ) ) )  =  (
(/)  u.  ( A  X.  ( B  \  C
) ) )
7 uncom 3633 . . 3  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( ( A  X.  ( B 
\  C ) )  u.  (/) )
8 un0 3796 . . 3  |-  ( ( A  X.  ( B 
\  C ) )  u.  (/) )  =  ( A  X.  ( B 
\  C ) )
97, 8eqtri 2472 . 2  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( A  X.  ( B  \  C ) )
101, 6, 93eqtrri 2477 1  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    \ cdif 3458    u. cun 3459   (/)c0 3770    X. cxp 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-xp 4995  df-rel 4996
This theorem is referenced by:  imadifxp  27436  sxbrsigalem2  28235
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