Theorem unixpss 4741

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 4740	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3834	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4219	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3191	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3834	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4219	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3191	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3129   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   X. cxp 4626

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-opab 4067  df-xp 4634

This theorem is referenced by:  relfld  5159