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Theorem difxp1 5371
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 5370 . 2  |-  ( ( A  X.  C ) 
\  ( B  X.  C ) )  =  ( ( ( A 
\  B )  X.  C )  u.  ( A  X.  ( C  \  C ) ) )
2 difid 3839 . . . . 5  |-  ( C 
\  C )  =  (/)
32xpeq2i 4963 . . . 4  |-  ( A  X.  ( C  \  C ) )  =  ( A  X.  (/) )
4 xp0 5364 . . . 4  |-  ( A  X.  (/) )  =  (/)
53, 4eqtri 2431 . . 3  |-  ( A  X.  ( C  \  C ) )  =  (/)
65uneq2i 3593 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  ( A  X.  ( C  \  C ) ) )  =  ( ( ( A  \  B )  X.  C
)  u.  (/) )
7 un0 3763 . 2  |-  ( ( ( A  \  B
)  X.  C )  u.  (/) )  =  ( ( A  \  B
)  X.  C )
81, 6, 73eqtrri 2436 1  |-  ( ( A  \  B )  X.  C )  =  ( ( A  X.  C )  \  ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    \ cdif 3410    u. cun 3411   (/)c0 3737    X. cxp 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-cnv 4950
This theorem is referenced by:  sxbrsigalem2  28614
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