Theorem unixpss 4792

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 4791	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3875	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4265	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3228	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3875	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4265	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3228	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3165   C_ wss 3167   ~Pcpw 3617   U.cuni 3852   X. cxp 4677

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-opab 4110  df-xp 4685

This theorem is referenced by:  relfld  5216