Theorem pwin 4379

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 3429	. . . 4 |- ( ( x C_ A /\ x C_ B ) <-> x C_ ( A i^i B ) )
2		vex 2805	. . . . . 6 |- x e. _V
3	2	elpw 3658	. . . . 5 |- ( x e. ~P A <-> x C_ A )
4	2	elpw 3658	. . . . 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 3658	. . . 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 3400	. 2 |- ( ~P A i^i ~P B ) = ~P ( A i^i B )
9	8	eqcomi 2235	1 |- ~P ( A i^i B ) = ( ~P A i^i ~P B )

Syntax hints:   /\ wa 104   = wceq 1397   e. wcel 2202   i^i cin 3199   C_ wss 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by: (None)