Theorem pwunss 4040

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 3152	. . 3 |- ( ( x C_ A \/ x C_ B ) -> x C_ ( A u. B ) )
2		elun 3114	. . . 4 |- ( x e. ( ~P A u. ~P B ) <-> ( x e. ~P A \/ x e. ~P B ) )
3		vex 2605	. . . . . 6 |- x e. _V
4	3	elpw 3390	. . . . 5 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3390	. . . . 5 |- ( x e. ~P B <-> x C_ B )
6	4, 5	orbi12i 714	. . . 4 |- ( ( x e. ~P A \/ x e. ~P B ) <-> ( x C_ A \/ x C_ B ) )
7	2, 6	bitri 182	. . 3 |- ( x e. ( ~P A u. ~P B ) <-> ( x C_ A \/ x C_ B ) )
8	3	elpw 3390	. . 3 |- ( x e. ~P ( A u. B ) <-> x C_ ( A u. B ) )
9	1, 7, 8	3imtr4i 199	. 2 |- ( x e. ( ~P A u. ~P B ) -> x e. ~P ( A u. B ) )
10	9	ssriv 3004	1 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Syntax hints:   \/ wo 662   e. wcel 1434   u. cun 2972   C_ wss 2974   ~Pcpw 3384

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386

This theorem is referenced by:  pwundifss  4042  pwunim  4043