Theorem unixpss 4776

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 4775	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3862	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 4250	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3217	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3862	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 4250	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3217	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 3155   C_ wss 3157   ~Pcpw 3605   U.cuni 3839   X. cxp 4661

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-opab 4095  df-xp 4669

This theorem is referenced by:  relfld  5198