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Metamath Proof Explorer

Theorem eq0ALT

Description: Alternate proof of eq0 . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion eq0ALT 
|- ( A = (/) <-> A. x -. x e. A )

Proof

Step Hyp Ref Expression
1 dfcleq 
 |-  ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
2 noel 
 |-  -. x e. (/)
3 2 nbn 
 |-  ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
4 3 albii 
 |-  ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) )
5 1 4 bitr4i 
 |-  ( A = (/) <-> A. x -. x e. A )