Theorem pwundifss 4375

Description: Break up the power class of a union into a union of smaller classes. (Contributed by Jim Kingdon, 30-Sep-2018.)

Assertion

Ref	Expression
pwundifss	|- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Proof of Theorem pwundifss

Step	Hyp	Ref	Expression
1		undif1ss 3566	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ( ~P ( A u. B ) u. ~P A )
2		pwunss 4373	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3378	. . . . 5 |- ( ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) ) <-> ( ~P A u. ~P B ) C_ ~P ( A u. B ) )
4	2, 3	mpbir 146	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 111	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3377	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 145	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	sseqtri 3258	1 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Syntax hints:   /\ wa 104   = wceq 1395   \ cdif 3194   u. cun 3195   C_ wss 3197   ~Pcpw 3649

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by: (None)