Theorem pwundifss 4246

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 3468	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ( ~P ( A u. B ) u. ~P A )
2		pwunss 4244	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3281	. . . . 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 145	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 110	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3280	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 144	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	sseqtri 3162	1 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Syntax hints:   /\ wa 103   = wceq 1335   \ cdif 3099   u. cun 3100   C_ wss 3102   ~Pcpw 3543

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545

This theorem is referenced by: (None)