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Metamath Proof Explorer

Theorem elpwd

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypotheses elpwd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
elpwd.2 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
Assertion elpwd ( 𝜑 → 𝐴 ∈ 𝒫 𝐵 )

Proof

Step Hyp Ref Expression
1 elpwd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2 elpwd.2 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
3 elpwg ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
4 1 3 syl ⊢ ( 𝜑 → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
5 2 4 mpbird ⊢ ( 𝜑 → 𝐴 ∈ 𝒫 𝐵 )