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 Gino Giotto and Steven Nguyen, 28-Jun-2024)

Ref Expression
Assertion eq0 A = x ¬ x A

Proof

Step Hyp Ref Expression
1 dfcleq A = x x A x
2 noel ¬ x
3 2 nbn ¬ x A x A x
4 3 albii x ¬ x A x x A x
5 1 4 bitr4i A = x ¬ x A