Theorem pwin 1915

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 1659	. . . 4 |- ((x (_ A /\ x (_ B) <-> x (_ (A i^i B))
2		visset 1350	. . . . . 6 |- x e. V
3	2	elpw 1801	. . . . 5 |- (x e. P~A <-> x (_ A)
4	2	elpw 1801	. . . . 5 |- (x e. P~B <-> x (_ B)
5	3, 4	anbi12i 369	. . . 4 |- ((x e. P~A /\ x e. P~B) <-> (x (_ A /\ x (_ B))
6	2	elpw 1801	. . . 4 |- (x e. P~(A i^i B) <-> x (_ (A i^i B))
7	1, 5, 6	3bitr4 158	. . 3 |- ((x e. P~A /\ x e. P~B) <-> x e. P~(A i^i B))
8	7	ineqri 1637	. 2 |- (P~A i^i P~B) = P~(A i^i B)
9	8	cleqcomi 1105	1 |- P~(A i^i B) = (P~A i^i P~B)

Syntax hints:   /\ wa 196   = wceq 1091   e. wcel 1092   i^i cin 1486   (_ wss 1487  P~cpw 1798

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799

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