Theorem elint2

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999)

	Ref	Expression
Hypothesis	elint2.1	⊢ 𝐴 ∈ V
Assertion	elint2	⊢ ( 𝐴 ∈ ∩ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		elint2.1	⊢ 𝐴 ∈ V
2	1	elint	⊢ ( 𝐴 ∈ ∩ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥 ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥 ) )
4	2 3	bitr4i	⊢ ( 𝐴 ∈ ∩ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 )