Metamath Proof Explorer


Theorem neq0

Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. (Contributed by NM, 21-Jun-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto, 28-Jun-2024)

Ref Expression
Assertion neq0
|- ( -. A = (/) <-> E. x x e. A )

Proof

Step Hyp Ref Expression
1 df-ex
 |-  ( E. x x e. A <-> -. A. x -. x e. A )
2 eq0
 |-  ( A = (/) <-> A. x -. x e. A )
3 1 2 xchbinxr
 |-  ( E. x x e. A <-> -. A = (/) )
4 3 bicomi
 |-  ( -. A = (/) <-> E. x x e. A )