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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwgded | Structured version Visualization version GIF version |
Description: elpwgdedVD 42537 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 4536 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 478 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 𝒫 cpw 4533 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-in 3894 df-ss 3904 df-pw 4535 |
This theorem is referenced by: sspwimp 42538 sspwimpALT 42545 |
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