Theorem clel4g

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 . (Revised by BJ, 1-Sep-2024)

	Ref	Expression
Assertion	clel4g	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elisset	⊢ ( 𝐵 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐵 )
2		biimt	⊢ ( ∃ 𝑥 𝑥 = 𝐵 → ( 𝐴 ∈ 𝐵 ↔ ( ∃ 𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ) )
3	1 2	syl	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ( ∃ 𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ) )
4		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) )
5	3 4	bitr4di	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ) )
6		eleq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵 ) )
7	6	bicomd	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥 ) )
8	7	pm5.74i	⊢ ( ( 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ↔ ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )
9	8	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )
10	5 9	bitrdi	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) ) )