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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwgded | Structured version Visualization version GIF version |
Description: elpwgdedVD 41244 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 4545 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 3 | biimpar 480 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵) |
5 | 1, 2, 4 | syl2an 597 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3943 df-ss 3952 df-pw 4541 |
This theorem is referenced by: sspwimp 41245 sspwimpALT 41252 |
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