Theorem unixpss 4469

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 4468	. . . . 5 |- ( A X. B ) C_ ~P ~P ( A u. B )
2	1	unissi 3624	. . . 4 |- U. ( A X. B ) C_ U. ~P ~P ( A u. B )
3		unipw 3972	. . . 4 |- U. ~P ~P ( A u. B ) = ~P ( A u. B )
4	2, 3	sseqtri 3031	. . 3 |- U. ( A X. B ) C_ ~P ( A u. B )
5	4	unissi 3624	. 2 |- U. U. ( A X. B ) C_ U. ~P ( A u. B )
6		unipw 3972	. 2 |- U. ~P ( A u. B ) = ( A u. B )
7	5, 6	sseqtri 3031	1 |- U. U. ( A X. B ) C_ ( A u. B )

Syntax hints:   u. cun 2971   C_ wss 2973   ~Pcpw 3382   U.cuni 3601   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-opab 3840  df-xp 4369

This theorem is referenced by:  relfld  4866