Theorem unixpss 4717

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 4716	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3812	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4195	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3176	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3812	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4195	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3176	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3114   C_ wss 3116   ~Pcpw 3559   U.cuni 3789   X. cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-opab 4044  df-xp 4610

This theorem is referenced by:  relfld  5132