Theorem clel4

Description: An alternate definition of class membership when the class is 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		eleq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵 ) )
3	1 2	ceqsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) ↔ 𝐴 ∈ 𝐵 )
4	3	bicomi	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )