Theorem unixpss 4863

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 4862	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3937	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4333	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3272	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3937	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4333	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3272	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3209   C_ wss 3211   ~Pcpw 3669   U.cuni 3914   X. cxp 4747

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-opab 4172  df-xp 4755

This theorem is referenced by:  relfld  5291