Theorem dfnul4

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 3-Sep-2024)

	Ref	Expression
Assertion	dfnul4	$$⊢\varnothing =\left\{x|\perp \right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfnul2	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$
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	eqtri	$$⊢\varnothing =\left\{x|\perp \right\}$$