Theorem abex

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd . (Contributed by AV, 19-Apr-2025)

	Ref	Expression
Hypotheses	abex.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
	abex.2	⊢ 𝐴 ∈ V
Assertion	abex	⊢ { 𝑥 ∣ 𝜑 } ∈ V

Proof

Step	Hyp	Ref	Expression
1		abex.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
2		abex.2	⊢ 𝐴 ∈ V
3		abss	⊢ ( { 𝑥 ∣ 𝜑 } ⊆ 𝐴 ↔ ∀ 𝑥 ( 𝜑 → 𝑥 ∈ 𝐴 ) )
4	3 1	mpgbir	⊢ { 𝑥 ∣ 𝜑 } ⊆ 𝐴
5	2 4	ssexi	⊢ { 𝑥 ∣ 𝜑 } ∈ V