Theorem noel

Description: The empty set has no elements. Theorem 6.14 of Quine p. 44. (Contributed by NM, 21-Jun-1993) (Proof shortened by Mario Carneiro, 1-Sep-2015) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024) (Proof shortened by BJ, 11-Oct-2026)

	Ref	Expression
Assertion	noel	|- -. A e. (/)

Proof

Step	Hyp	Ref	Expression
1		fal	|- -. F.
2		dfnul4	|- (/) = { y | F. }
3	2	eleq2i	|- ( x e. (/) <-> x e. { y | F. } )
4		df-clab	|- ( x e. { y | F. } <-> [ x / y ] F. )
5		sbv	|- ( [ x / y ] F. <-> F. )
6	3 4 5	3bitri	|- ( x e. (/) <-> F. )
7	1 6	mtbir	|- -. x e. (/)
8	7	nex	|- -. E. x x e. (/)
9		elex2	|- ( A e. (/) -> E. x x e. (/) )
10	8 9	mto	|- -. A e. (/)