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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwgded | Structured version Visualization version GIF version |
Description: elpwgdedVD 39902 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elpwgded.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
elpwgded.2 | ⊢ (𝜓 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
elpwgded | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwgded.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | elpwgded.2 | . 2 ⊢ (𝜓 → 𝐴 ⊆ 𝐵) | |
3 | elpwg 4356 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 470 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | syl2an 590 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 Vcvv 3384 ⊆ wss 3768 𝒫 cpw 4348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2776 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-v 3386 df-in 3775 df-ss 3782 df-pw 4350 |
This theorem is referenced by: sspwimp 39903 sspwimpALT 39910 |
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