Theorem clel4

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypothesis	clel4.1	⊢ 𝐵 ∈ V
Assertion	clel4	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		clel4.1	⊢ 𝐵 ∈ V
2		clel4g	⊢ ( 𝐵 ∈ V → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )