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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 4603. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 44584 is elpwgdedVD 44937 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 4603 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
| 4 | 3 | biimpar 477 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
| 5 | 1, 2, 4 | el12 44746 | 1 ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ( wvd1 44589 ( wvhc2 44600 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-pw 4602 df-vd1 44590 df-vhc2 44601 |
| This theorem is referenced by: sspwimpVD 44939 |
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