Metamath Proof Explorer

Theorem nulmo

Description: There exists at most one empty set. With either axnul or axnulALT or ax-nul , this proves that there exists a unique empty set. In practice, once the language of classes is available, we use the stronger characterization among classes eq0 . (Contributed by NM, 22-Dec-2007) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022) (Proof shortened by Wolf Lammen, 26-Apr-2023)

Ref Expression
Assertion nulmo * x y ¬ y x


Step Hyp Ref Expression
1 nfv x
2 1 axextmo * x y y x
3 nbfal ¬ y x y x
4 3 albii y ¬ y x y y x
5 4 mobii * x y ¬ y x * x y y x
6 2 5 mpbir * x y ¬ y x