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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 4116. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 37801 is elpwgdedVD 38175 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 4116 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 501 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | el12 37974 | 1 ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ( wvd1 37806 ( wvhc2 37817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-vd1 37807 df-vhc2 37818 |
This theorem is referenced by: sspwimpVD 38177 |
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