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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpwd | Structured version Visualization version GIF version |
Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
elpwd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elpwd.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
elpwd | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwd.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | elpwd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | elpwg 4116 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
5 | 1, 4 | mpbird 246 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 |
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 |
This theorem is referenced by: dmvolss 38878 sge0xaddlem1 39326 ovnval2b 39442 ovnsubadd2lem 39535 vonvolmbllem 39550 vonvolmbl 39551 smfresal 39673 smfpimbor1lem1 39683 |
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