Theorem bj-elpwg 34372

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 5244 (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 4545	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		ssidd 3987	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴)
3		id 22	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
4	2, 3	ssind 4206	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ∩ 𝐵))
5		ssexg 5224	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
6	4, 5	sylan 582	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
7		elpwg 4539	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
8	7	biimparc 482	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
9	6, 8	syldan 593	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ 𝒫 𝐵)
10	9	expcom 416	. 2 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝒫 𝐵))
11	1, 10	impbid2 228	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2113  Vcvv 3493   ∩ cin 3932   ⊆ wss 3933  𝒫 cpw 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3495  df-in 3940  df-ss 3949  df-pw 4538

This theorem is referenced by: (None)