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	|- E* x A. y -. y e. x

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ x F.
2	1	axextmo	|- E* x A. y ( y e. x <-> F. )
3		nbfal	|- ( -. y e. x <-> ( y e. x <-> F. ) )
4	3	albii	|- ( A. y -. y e. x <-> A. y ( y e. x <-> F. ) )
5	4	mobii	|- ( E* x A. y -. y e. x <-> E* x A. y ( y e. x <-> F. ) )
6	2 5	mpbir	|- E* x A. y -. y e. x