Theorem difxp2 5560

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

Step	Hyp	Ref	Expression
1		difxp 5558	. 2 |- ( ( A X. B ) \ ( A X. C ) ) = ( ( ( A \ A ) X. B ) u. ( A X. ( B \ C ) ) )
2		difid 3948	. . . . 5 |- ( A \ A ) = (/)
3	2	xpeq1i 5135	. . . 4 |- ( ( A \ A ) X. B ) = ( (/) X. B )
4		0xp 5199	. . . 4 |- ( (/) X. B ) = (/)
5	3, 4	eqtri 2644	. . 3 |- ( ( A \ A ) X. B ) = (/)
6	5	uneq1i 3763	. 2 |- ( ( ( A \ A ) X. B ) u. ( A X. ( B \ C ) ) ) = ( (/) u. ( A X. ( B \ C ) ) )
7		uncom 3757	. . 3 |- ( (/) u. ( A X. ( B \ C ) ) ) = ( ( A X. ( B \ C ) ) u. (/) )
8		un0 3967	. . 3 |- ( ( A X. ( B \ C ) ) u. (/) ) = ( A X. ( B \ C ) )
9	7, 8	eqtri 2644	. 2 |- ( (/) u. ( A X. ( B \ C ) ) ) = ( A X. ( B \ C ) )
10	1, 6, 9	3eqtrri 2649	1 |- ( A X. ( B \ C ) ) = ( ( A X. B ) \ ( A X. C ) )

Syntax hints:   = wceq 1483   \ cdif 3571   u. cun 3572   (/)c0 3915   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  imadifxp  29414  sxbrsigalem2  30348