Theorem unixpss 5234

Description: The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
unixpss	|- U. U. ( A X. B ) C_ ( A u. B )

Proof of Theorem unixpss

Step	Hyp	Ref	Expression
1		xpsspw 5233	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 4461	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4918	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3637	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 4461	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4918	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3637	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3572   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  relfld  5661  filnetlem3  32375