Theorem pwin 4333

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 3396	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
2		vex 2776	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	elpw 3623	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	2	elpw 3623	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	3, 4	anbi12i 460	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))
6	2	elpw 3623	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
7	1, 5, 6	3bitr4i 212	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵))
8	7	ineqri 3367	. 2 ⊢ (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴 ∩ 𝐵)
9	8	eqcomi 2210	1 ⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ∩ cin 3166   ⊆ wss 3167  𝒫 cpw 3617

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3173  df-ss 3180  df-pw 3619

This theorem is referenced by: (None)