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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwgdedVD | Structured version Visualization version GIF version |
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4605. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 43315 is elpwgdedVD 43668 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpwgdedVD.1 | ⊢ ( 𝜑 ▶ 𝐴 ∈ V ) |
elpwgdedVD.2 | ⊢ ( 𝜓 ▶ 𝐴 ⊆ 𝐵 ) |
Ref | Expression |
---|---|
elpwgdedVD | ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwgdedVD.1 | . 2 ⊢ ( 𝜑 ▶ 𝐴 ∈ V ) | |
2 | elpwgdedVD.2 | . 2 ⊢ ( 𝜓 ▶ 𝐴 ⊆ 𝐵 ) | |
3 | elpwg 4605 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 478 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | el12 43477 | 1 ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 𝒫 cpw 4602 ( wvd1 43320 ( wvhc2 43331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-pw 4604 df-vd1 43321 df-vhc2 43332 |
This theorem is referenced by: sspwimpVD 43670 |
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