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	$$⊢\phi \to \neg x\in A$$
Assertion	eq0rdv	$$⊢\phi \to A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		eq0rdv.1	$$⊢\phi \to \neg x\in A$$
2	1	alrimiv	$$⊢\phi \to \forall x\neg x\in A$$
3		eq0	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$
4	2 3	sylibr	$$⊢\phi \to A=\varnothing$$