Theorem bj-elpwg 37018

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 4625 and elpw2g 5351 (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 4629	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		ssidd 4032	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴)
3		id 22	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4	2, 3	ssind 4262	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ∩ 𝐵))
5		ssexg 5341	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
6	4, 5	sylan 579	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
7		elpwg 4625	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
8	7	biimparc 479	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
9	6, 8	syldan 590	. . 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 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624

This theorem is referenced by: (None)