Theorem clel2

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypothesis	clel2.1	⊢ 𝐴 ∈ V
Assertion	clel2	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		clel2.1	⊢ 𝐴 ∈ V
2		clel2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )