Theorem pwin 4254

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 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)

Proof of Theorem pwin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssin 3339	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
2		vex 2724	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpw 3559	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	2	elpw 3559	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	3, 4	anbi12i 456	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))
6	2	elpw 3559	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
7	1, 5, 6	3bitr4i 211	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵))
8	7	ineqri 3310	. 2 ⊢ (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴 ∩ 𝐵)
9	8	eqcomi 2168	1 ⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)

Syntax hints:   ∧ wa 103   = wceq 1342   ∈ wcel 2135   ∩ cin 3110   ⊆ wss 3111  𝒫 cpw 3553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124  df-pw 3555

This theorem is referenced by: (None)