Theorem pwundifss 4332

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 3535	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ( ~P ( A u. B ) u. ~P A )
2		pwunss 4330	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3347	. . . . 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 3346	. . 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 3227	1 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) C_ ~P ( A u. B )

Syntax hints:   /\ wa 104   = wceq 1373   \ cdif 3163   u. cun 3164   C_ wss 3166   ~Pcpw 3616

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618

This theorem is referenced by: (None)