Theorem unixpss 4772

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 4771	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3858	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4246	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3213	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3858	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4246	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3213	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3151   C_ wss 3153   ~Pcpw 3601   U.cuni 3835   X. cxp 4657

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 4147  ax-pow 4203

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 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-opab 4091  df-xp 4665

This theorem is referenced by:  relfld  5194