Theorem pwundif 4730

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 3847	. 2 |- ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) = ( ~P ( A u. B ) u. ~P A )
2		pwunss 4728	. . . . 5 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )
3		unss 3617	. . . . 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 456	. . 3 |- ~P A C_ ~P ( A u. B )
6		ssequn2 3616	. . 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 2432	1 |- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A )

Syntax hints:   /\ wa 367   = wceq 1405   \ cdif 3411   u. cun 3412   C_ wss 3414   ~Pcpw 3955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-pw 3957

This theorem is referenced by:  pwfilem  7848