Theorem pwin 4284

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 3359	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		vex 2742	. . . . . 6 |- x e. _V
3	2	elpw 3583	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4	2	elpw 3583	. . . . 5 |- ( x e. ~P B <-> x C_ B )
5	3, 4	anbi12i 460	. . . 4 |- ( ( x e. ~P A /\ x e. ~P B ) <-> ( x C_ A /\ x C_ B ) )
6	2	elpw 3583	. . . 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 3330	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
9	8	eqcomi 2181	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   i^i cin 3130   C_ wss 3131   ~Pcpw 3577

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by: (None)