Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009)
Ref | Expression | ||
---|---|---|---|
Assertion | pwss | ⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 | ⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) ) | |
2 | velpw | ⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) | |
3 | 2 | imbi1i | ⊢ ( ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
4 | 3 | albii | ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
5 | 1 4 | bitri | ⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) ) |