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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 4567. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 42938 is elpwgdedVD 43291 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 4567 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 479 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | el12 43100 | 1 ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 𝒫 cpw 4564 ( wvd1 42943 ( wvhc2 42954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 df-pw 4566 df-vd1 42944 df-vhc2 42955 |
This theorem is referenced by: sspwimpVD 43293 |
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