Theorem pwundif 4760

Description: Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)

Assertion

Ref	Expression
pwundif	|- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Proof of Theorem pwundif

Step	Hyp	Ref	Expression
1		undif1 3854	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) = ( ~P ( A u. B ) u. ~P A )
2		pwunss 4758	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3620	. . . . 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 214	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 464	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3619	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 213	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	eqtr2i 2485	1 |- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Syntax hints:   /\ wa 375   = wceq 1455   \ cdif 3413   u. cun 3414   C_ wss 3416   ~Pcpw 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-pw 3965

This theorem is referenced by:  pwfilem  7894