Theorem eq0rdv

Description: Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

	Ref	Expression
Hypothesis	eq0rdv.1	⊢ ( 𝜑 → ¬ 𝑥 ∈ 𝐴 )
Assertion	eq0rdv	⊢ ( 𝜑 → 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		eq0rdv.1	⊢ ( 𝜑 → ¬ 𝑥 ∈ 𝐴 )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
3		eq0	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
4	2 3	sylibr	⊢ ( 𝜑 → 𝐴 = ∅ )