Theorem bj-elpwg 37253

Description: If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4557 and elpw2g 5278 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)

Assertion

Ref	Expression
bj-elpwg	⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Proof of Theorem bj-elpwg

Step	Hyp	Ref	Expression
1		elpwi 4561	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		ssidd 3957	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴)
3		id 22	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4	2, 3	ssind 4193	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ∩ 𝐵))
5		ssexg 5268	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
6	4, 5	sylan 580	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
7		elpwg 4557	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
8	7	biimparc 479	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
9	6, 8	syldan 591	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ 𝒫 𝐵)
10	9	expcom 413	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝒫 𝐵))
11	1, 10	impbid2 226	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-in 3908  df-ss 3918  df-pw 4556

This theorem is referenced by: (None)