Theorem pwin 5018

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

Assertion

Ref	Expression
pwin	|- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Proof of Theorem pwin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssin 3835	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		selpw 4165	. . . . 5 |- ( x e. ~P A <-> x C_ A )
3		selpw 4165	. . . . 5 |- ( x e. ~P B <-> x C_ B )
4	2, 3	anbi12i 733	. . . 4 |- ( ( x e. ~P A /\ x e. ~P B ) <-> ( x C_ A /\ x C_ B ) )
5		selpw 4165	. . . 4 |- ( x e. ~P ( A i^i B ) <-> x C_ ( A i^i B ) )
6	1, 4, 5	3bitr4i 292	. . 3 |- ( ( x e. ~P A /\ x e. ~P B ) <-> x e. ~P ( A i^i B ) )
7	6	ineqri 3806	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
8	7	eqcomi 2631	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   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-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by: (None)