Theorem elintabg

Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by NM, 30-Aug-1993) Put in closed form. (Revised by RP, 13-Aug-2020)

	Ref	Expression
Assertion	elintabg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elintg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑦 ∈ { 𝑥 ∣ 𝜑 } 𝐴 ∈ 𝑦 ) )
2		eleq2w	⊢ ( 𝑦 = 𝑥 → ( 𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥 ) )
3	2	ralab2	⊢ ( ∀ 𝑦 ∈ { 𝑥 ∣ 𝜑 } 𝐴 ∈ 𝑦 ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) )
4	1 3	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )