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 = x x A

Proof

Step Hyp Ref Expression
1 df-ex x x A ¬ x ¬ x A
2 eq0 A = x ¬ x A
3 1 2 xchbinxr x x A ¬ A =
4 3 bicomi ¬ A = x x A