Theorem pwin 4254

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 3339	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		vex 2724	. . . . . 6 |- x e. _V
3	2	elpw 3559	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4	2	elpw 3559	. . . . 5 |- ( x e. ~P B <-> x C_ B )
5	3, 4	anbi12i 456	. . . 4 |- ( ( x e. ~P A /\ x e. ~P B ) <-> ( x C_ A /\ x C_ B ) )
6	2	elpw 3559	. . . 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 3310	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
9	8	eqcomi 2168	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 103   = wceq 1342   e. wcel 2135   i^i cin 3110   C_ wss 3111   ~Pcpw 3553

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124  df-pw 3555

This theorem is referenced by: (None)