Theorem unixpss 4837

Description: The double class union of a cross 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 4836	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3914	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4307	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3259	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3914	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4307	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3259	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3196   C_ wss 3198   ~Pcpw 3650   U.cuni 3891   X. cxp 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-opab 4149  df-xp 4729

This theorem is referenced by:  relfld  5263