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Metamath Proof Explorer

Theorem eq0

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

Ref Expression
Assertion eq0 
|- ( A = (/) <-> A. x -. x e. A )

Proof

Step Hyp Ref Expression
1 biidd 
 |-  ( y = x -> ( F. <-> F. ) )
2 1 eqabbw 
 |-  ( A = { y | F. } <-> A. x ( x e. A <-> F. ) )
3 dfnul4 
 |-  (/) = { y | F. }
4 3 eqeq2i 
 |-  ( A = (/) <-> A = { y | F. } )
5 nbfal 
 |-  ( -. x e. A <-> ( x e. A <-> F. ) )
6 5 albii 
 |-  ( A. x -. x e. A <-> A. x ( x e. A <-> F. ) )
7 2 4 6 3bitr4i 
 |-  ( A = (/) <-> A. x -. x e. A )