Theorem pwin 4199

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 3293	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		vex 2684	. . . . . 6 |- x e. _V
3	2	elpw 3511	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4	2	elpw 3511	. . . . 5 |- ( x e. ~P B <-> x C_ B )
5	3, 4	anbi12i 455	. . . 4 |- ( ( x e. ~P A /\ x e. ~P B ) <-> ( x C_ A /\ x C_ B ) )
6	2	elpw 3511	. . . 4 |- ( x e. ~P ( A i^i B ) <-> x C_ ( A i^i B ) )
7	1, 5, 6	3bitr4i 211	. . 3 |- ( ( x e. ~P A /\ x e. ~P B ) <-> x e. ~P ( A i^i B ) )
8	7	ineqri 3264	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
9	8	eqcomi 2141	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   i^i cin 3065   C_ wss 3066   ~Pcpw 3505

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507

This theorem is referenced by: (None)