Theorem dfnul2

Description: Alternate definition of the empty set. Definition 5.14 of TakeutiZaring p. 20. (Contributed by NM, 26-Dec-1996) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul2	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfnul4	$$⊢\varnothing =\left\{x|\perp \right\}$$
2		equid	$$⊢x=x$$
3	2	notnoti	$$⊢\neg \neg x=x$$
4	3	bifal	$$⊢\neg x=x\leftrightarrow \perp$$
5	4	abbii	$$⊢\left\{x|\neg x=x\right\}=\left\{x|\perp \right\}$$
6	1 5	eqtr4i	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$