This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 
|- ( ph -> -. x e. A )
Assertion eq0rdv 
|- ( ph -> A = (/) )

Proof

Step Hyp Ref Expression
1 eq0rdv.1 
 |-  ( ph -> -. x e. A )
2 1 alrimiv 
 |-  ( ph -> A. x -. x e. A )
3 eq0 
 |-  ( A = (/) <-> A. x -. x e. A )
4 2 3 sylibr 
 |-  ( ph -> A = (/) )