Theorem pwunss 4163

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 3219	. . 3 |- ( ( x C_ A \/ x C_ B ) -> x C_ ( A u. B ) )
2		elun 3181	. . . 4 |- ( x e. ( ~P A u. ~P B ) <-> ( x e. ~P A \/ x e. ~P B ) )
3		vex 2658	. . . . . 6 |- x e. _V
4	3	elpw 3480	. . . . 5 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3480	. . . . 5 |- ( x e. ~P B <-> x C_ B )
6	4, 5	orbi12i 736	. . . 4 |- ( ( x e. ~P A \/ x e. ~P B ) <-> ( x C_ A \/ x C_ B ) )
7	2, 6	bitri 183	. . 3 |- ( x e. ( ~P A u. ~P B ) <-> ( x C_ A \/ x C_ B ) )
8	3	elpw 3480	. . 3 |- ( x e. ~P ( A u. B ) <-> x C_ ( A u. B ) )
9	1, 7, 8	3imtr4i 200	. 2 |- ( x e. ( ~P A u. ~P B ) -> x e. ~P ( A u. B ) )
10	9	ssriv 3065	1 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Syntax hints:   \/ wo 680   e. wcel 1461   u. cun 3033   C_ wss 3035   ~Pcpw 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476

This theorem is referenced by:  pwundifss  4165  pwunim  4166