Theorem pwunss 4205

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 3255	. . 3 |- ( ( x C_ A \/ x C_ B ) -> x C_ ( A u. B ) )
2		elun 3217	. . . 4 |- ( x e. ( ~P A u. ~P B ) <-> ( x e. ~P A \/ x e. ~P B ) )
3		vex 2689	. . . . . 6 |- x e. _V
4	3	elpw 3516	. . . . 5 |- ( x e. ~P A <-> x C_ A )
5	3	elpw 3516	. . . . 5 |- ( x e. ~P B <-> x C_ B )
6	4, 5	orbi12i 753	. . . 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 3516	. . 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 3101	1 |- ( ~P A u. ~P B ) C_ ~P ( A u. B )

Syntax hints:   \/ wo 697   e. wcel 1480   u. cun 3069   C_ wss 3071   ~Pcpw 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512

This theorem is referenced by:  pwundifss  4207  pwunim  4208