Theorem pwundif 4739

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 3865	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) = ( ~P ( A u. B ) u. ~P A )
2		pwunss 4737	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3641	. . . . 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 209	. . . 4 |- ( ~P A C_ ~P ( A u. B ) /\ ~P B C_ ~P ( A u. B ) )
5	4	simpli 458	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3640	. . 3 |- ( ~P A C_ ~P ( A u. B ) <-> ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B ) )
7	5, 6	mpbi 208	. 2 |- ( ~P ( A u. B ) u. ~P A ) = ~P ( A u. B )
8	1, 7	eqtr2i 2484	1 |- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Syntax hints:   /\ wa 369   = wceq 1370   \ cdif 3436   u. cun 3437   C_ wss 3439   ~Pcpw 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-pw 3973

This theorem is referenced by:  pwfilem  7719