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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-elpwg | Structured version Visualization version GIF version |
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 4605 and elpw2g 5344 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.) |
Ref | Expression |
---|---|
bj-elpwg | ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4609 | . 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssidd 4005 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴) | |
3 | id 22 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
4 | 2, 3 | ssind 4232 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ∩ 𝐵)) |
5 | ssexg 5323 | . . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V) | |
6 | 4, 5 | sylan 579 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ∈ 𝑉) → 𝐴 ∈ V) |
7 | elpwg 4605 | . . . . 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 225 | 1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 df-pw 4604 |
This theorem is referenced by: (None) |
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