Theorem unixpss 4724

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 4723	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3819	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4202	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3181	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3819	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4202	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3181	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3119   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-opab 4051  df-xp 4617

This theorem is referenced by:  relfld  5139