This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ssab

Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006)

Ref Expression
Assertion ssab ( 𝐴 ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 abid2 ⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = 𝐴
2 1 sseq1i ⊢ ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } ⊆ { 𝑥 ∣ 𝜑 } ↔ 𝐴 ⊆ { 𝑥 ∣ 𝜑 } )
3 ss2ab ⊢ ( { 𝑥 ∣ 𝑥 ∈ 𝐴 } ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4 2 3 bitr3i ⊢ ( 𝐴 ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )