Theorem unixpss 4883

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 4882	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3953	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4352	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3282	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3953	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4352	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3282	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3218   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-opab 4188  df-xp 4775

This theorem is referenced by:  relfld  5311