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Metamath Proof Explorer

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 = ∅ )