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Theorem difxp2 5269
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 5267 . 2  |-  ( ( A  X.  B ) 
\  ( A  X.  C ) )  =  ( ( ( A 
\  A )  X.  B )  u.  ( A  X.  ( B  \  C ) ) )
2 difid 3747 . . . . 5  |-  ( A 
\  A )  =  (/)
32xpeq1i 4859 . . . 4  |-  ( ( A  \  A )  X.  B )  =  ( (/)  X.  B
)
4 0xp 4920 . . . 4  |-  ( (/)  X.  B )  =  (/)
53, 4eqtri 2493 . . 3  |-  ( ( A  \  A )  X.  B )  =  (/)
65uneq1i 3575 . 2  |-  ( ( ( A  \  A
)  X.  B )  u.  ( A  X.  ( B  \  C ) ) )  =  (
(/)  u.  ( A  X.  ( B  \  C
) ) )
7 uncom 3569 . . 3  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( ( A  X.  ( B 
\  C ) )  u.  (/) )
8 un0 3762 . . 3  |-  ( ( A  X.  ( B 
\  C ) )  u.  (/) )  =  ( A  X.  ( B 
\  C ) )
97, 8eqtri 2493 . 2  |-  ( (/)  u.  ( A  X.  ( B  \  C ) ) )  =  ( A  X.  ( B  \  C ) )
101, 6, 93eqtrri 2498 1  |-  ( A  X.  ( B  \  C ) )  =  ( ( A  X.  B )  \  ( A  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    \ cdif 3387    u. cun 3388   (/)c0 3722    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-rel 4846
This theorem is referenced by:  imadifxp  28288  sxbrsigalem2  29181
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