Theorem unixpss 4845

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 4844	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3921	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4315	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3262	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3921	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4315	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3262	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3199   C_ wss 3201   ~Pcpw 3656   U.cuni 3898   X. cxp 4729

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-opab 4156  df-xp 4737

This theorem is referenced by:  relfld  5272