Theorem unixpss 4773

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 4772	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3859	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4247	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3214	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3859	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4247	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3214	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3152   C_ wss 3154   ~Pcpw 3602   U.cuni 3836   X. cxp 4658

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-opab 4092  df-xp 4666

This theorem is referenced by:  relfld  5195