Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 31-Dec-1993) (Proof shortened by BJ, 31-Dec-2023)
Ref | Expression | ||
---|---|---|---|
Hypothesis | elpw.1 | ⊢ 𝐴 ∈ V | |
Assertion | elpw | ⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpw.1 | ⊢ 𝐴 ∈ V | |
2 | elpwg | ⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) ) | |
3 | 1 2 | ax-mp | ⊢ ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) |