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Theorem difxp2 5431
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 5429 . 2  |-  ( ( A  X.  B ) 
\  ( A  X.  C ) )  =  ( ( ( A 
\  A )  X.  B )  u.  ( A  X.  ( B  \  C ) ) )
2 difid 3895 . . . . 5  |-  ( A 
\  A )  =  (/)
32xpeq1i 5019 . . . 4  |-  ( ( A  \  A )  X.  B )  =  ( (/)  X.  B
)
4 0xp 5078 . . . 4  |-  ( (/)  X.  B )  =  (/)
53, 4eqtri 2496 . . 3  |-  ( ( A  \  A )  X.  B )  =  (/)
65uneq1i 3654 . 2  |-  ( ( ( A  \  A
)  X.  B )  u.  ( A  X.  ( B  \  C ) ) )  =  (
(/)  u.  ( A  X.  ( B  \  C
) ) )
7 uncom 3648 . . 3  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( ( A  X.  ( B 
\  C ) )  u.  (/) )
8 un0 3810 . . 3  |-  ( ( A  X.  ( B 
\  C ) )  u.  (/) )  =  ( A  X.  ( B 
\  C ) )
97, 8eqtri 2496 . 2  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( A  X.  ( B  \  C ) )
101, 6, 93eqtrri 2501 1  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    u. cun 3474   (/)c0 3785    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006
This theorem is referenced by:  imadifxp  27131  sxbrsigalem2  27897
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