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) Prove directly from definition to allow shortening dfnul2 . (Revised by BJ, 23-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		df-nul	$$⊢\varnothing =\mathrm{V}\setminus \mathrm{V}$$
2		df-dif	$$⊢\mathrm{V}\setminus \mathrm{V}=\left\{x|\left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)\right\}$$
3		pm3.24	$$⊢\neg \left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)$$
4	3	bifal	$$⊢\left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)\leftrightarrow \perp$$
5	4	abbii	$$⊢\left\{x|\left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)\right\}=\left\{x|\perp \right\}$$
6	1 2 5	3eqtri	$$⊢\varnothing =\left\{x|\perp \right\}$$