Theorem pwin 2831

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.

Assertion

Ref	Expression
pwin	|- P~(A i^i B) = (P~A i^i P~B)

Proof of Theorem pwin

Step	Hyp	Ref	Expression
1		ssin 2235	. . . 4 |- ((x (_ A /\ x (_ B) <-> x (_ (A i^i B))
2		visset 1816	. . . . . 6 |- x e. V
3	2	elpw 2408	. . . . 5 |- (x e. P~A <-> x (_ A)
4	2	elpw 2408	. . . . 5 |- (x e. P~B <-> x (_ B)
5	3, 4	anbi12i 484	. . . 4 |- ((x e. P~A /\ x e. P~B) <-> (x (_ A /\ x (_ B))
6	2	elpw 2408	. . . 4 |- (x e. P~(A i^i B) <-> x (_ (A i^i B))
7	1, 5, 6	3bitr4 183	. . 3 |- ((x e. P~A /\ x e. P~B) <-> x e. P~(A i^i B))
8	7	ineqri 2212	. 2 |- (P~A i^i P~B) = P~(A i^i B)
9	8	eqcomi 1482	1 |- P~(A i^i B) = (P~A i^i P~B)

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960   i^i cin 2049   (_ wss 2050  P~cpw 2405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406

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