Theorem pwin 4422

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 3453	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		vex 2824	. . . . . 6 |- x e. _V
3	2	elpw 3691	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4	2	elpw 3691	. . . . 5 |- ( x e. ~P B <-> x C_ B )
5	3, 4	anbi12i 464	. . . 4 |- ( ( x e. ~P A /\ x e. ~P B ) <-> ( x C_ A /\ x C_ B ) )
6	2	elpw 3691	. . . 4 |- ( x e. ~P ( A i^i B ) <-> x C_ ( A i^i B ) )
7	1, 5, 6	3bitr4i 212	. . 3 |- ( ( x e. ~P A /\ x e. ~P B ) <-> x e. ~P ( A i^i B ) )
8	7	ineqri 3424	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
9	8	eqcomi 2242	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   ~Pcpw 3685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by: (None)