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)

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

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		equid	$$⊢x=x$$
5	4	notnoti	$$⊢\neg \neg x=x$$
6	3 5	2false	$$⊢\left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)\leftrightarrow \neg x=x$$
7	6	abbii	$$⊢\left\{x|\left(x\in \mathrm{V}\wedge \neg x\in \mathrm{V}\right)\right\}=\left\{x|\neg x=x\right\}$$
8	1 2 7	3eqtri	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$