Theorem pwunss 5019

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
pwunss	|- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Proof of Theorem pwunss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun 3792	. . 3 |- ( ( x C_ A \/ x C_ B ) -> x C_ ( A u. B ) )
2		elun 3753	. . . 4 |- ( x e. ( ~P A u. ~P B ) <-> ( x e. ~P A \/ x e. ~P B ) )
3		selpw 4165	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4		selpw 4165	. . . . 5 |- ( x e. ~P B <-> x C_ B )
5	3, 4	orbi12i 543	. . . 4 |- ( ( x e. ~P A \/ x e. ~P B ) <-> ( x C_ A \/ x C_ B ) )
6	2, 5	bitri 264	. . 3 |- ( x e. ( ~P A u. ~P B ) <-> ( x C_ A \/ x C_ B ) )
7		selpw 4165	. . 3 |- ( x e. ~P ( A u. B ) <-> x C_ ( A u. B ) )
8	1, 6, 7	3imtr4i 281	. 2 |- ( x e. ( ~P A u. ~P B ) -> x e. ~P ( A u. B ) )
9	8	ssriv 3607	1 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Syntax hints:   \/ wo 383   e. wcel 1990   u. cun 3572   C_ wss 3574   ~Pcpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  pwundif  5021  pwun  5022