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| Mirrors > Home > MPE Home > Th. List > elpwg | Structured version Visualization version GIF version | ||
| Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5333. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.) |
| Ref | Expression |
|---|---|
| elpwg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq1 4009 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
| 2 | df-pw 4602 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
| 3 | 1, 2 | elab2g 3680 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
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