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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 4311. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 39301 is elpwgdedVD 39671 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 4311 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 503 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | el12 39474 | 1 ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2140 Vcvv 3341 ⊆ wss 3716 𝒫 cpw 4303 ( wvd1 39306 ( wvhc2 39317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-v 3343 df-in 3723 df-ss 3730 df-pw 4305 df-vd1 39307 df-vhc2 39318 |
This theorem is referenced by: sspwimpVD 39673 |
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