Theorem bj-elpwg 37412

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 4539 and elpw2g 5268 (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 4543	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		ssidd 3945	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴)
3		id 22	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4	2, 3	ssind 4176	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ∩ 𝐵))
5		ssexg 5258	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
6	4, 5	sylan 586	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
7		elpwg 4539	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
8	7	biimparc 480	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
9	6, 8	syldan 597	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ 𝒫 𝐵)
10	9	expcom 414	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝒫 𝐵))
11	1, 10	impbid2 227	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-in 3897  df-ss 3907  df-pw 4538

This theorem is referenced by: (None)