Theorem pwunss 4020

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 3122	. . 3 |- ( ( x C_ A \/ x C_ B ) -> x C_ ( A u. B ) )
2		elun 3084	. . . 4 |- ( x e. ( ~P A u. ~P B ) <-> ( x e. ~P A \/ x e. ~P B ) )
3		vex 2560	. . . . . 6 |- x e. _V
4	3	elpw 3365	. . . . 5 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3365	. . . . 5 |- ( x e. ~P B <-> x C_ B )
6	4, 5	orbi12i 681	. . . 4 |- ( ( x e. ~P A \/ x e. ~P B ) <-> ( x C_ A \/ x C_ B ) )
7	2, 6	bitri 173	. . 3 |- ( x e. ( ~P A u. ~P B ) <-> ( x C_ A \/ x C_ B ) )
8	3	elpw 3365	. . 3 |- ( x e. ~P ( A u. B ) <-> x C_ ( A u. B ) )
9	1, 7, 8	3imtr4i 190	. 2 |- ( x e. ( ~P A u. ~P B ) -> x e. ~P ( A u. B ) )
10	9	ssriv 2949	1 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Syntax hints:   \/ wo 629   e. wcel 1393   u. cun 2915   C_ wss 2917   ~Pcpw 3359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361

This theorem is referenced by:  pwundifss  4022  pwunim  4023