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	⊢ ℘(A ∩ B) = (℘A ∩ ℘B)

Proof of Theorem pwin

Step	Hyp	Ref	Expression
1		ssin 1659	. . . 4 ⊢ ((x ⊆ A ∧ x ⊆ B) ↔ x ⊆ (A ∩ B))
2		visset 1350	. . . . . 6 ⊢ x ∈ V
3	2	elpw 1801	. . . . 5 ⊢ (x ∈ ℘A ↔ x ⊆ A)
4	2	elpw 1801	. . . . 5 ⊢ (x ∈ ℘B ↔ x ⊆ B)
5	3, 4	anbi12i 369	. . . 4 ⊢ ((x ∈ ℘A ∧ x ∈ ℘B) ↔ (x ⊆ A ∧ x ⊆ B))
6	2	elpw 1801	. . . 4 ⊢ (x ∈ ℘(A ∩ B) ↔ x ⊆ (A ∩ B))
7	1, 5, 6	3bitr4 158	. . 3 ⊢ ((x ∈ ℘A ∧ x ∈ ℘B) ↔ x ∈ ℘(A ∩ B))
8	7	ineqri 1637	. 2 ⊢ (℘A ∩ ℘B) = ℘(A ∩ B)
9	8	cleqcomi 1105	1 ⊢ ℘(A ∩ B) = (℘A ∩ ℘B)

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  ℘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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