Theorem dfnul3

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul3	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		fal	$$⊢\neg \perp$$
2		pm3.24	$$⊢\neg \left(x\in A\wedge \neg x\in A\right)$$
3	1 2	2false	$$⊢\perp \leftrightarrow \left(x\in A\wedge \neg x\in A\right)$$
4	3	abbii	$$⊢\left\{x|\perp \right\}=\left\{x|\left(x\in A\wedge \neg x\in A\right)\right\}$$
5		dfnul4	$$⊢\varnothing =\left\{x|\perp \right\}$$
6		df-rab	$$⊢\left\{x\in A|\neg x\in A\right\}=\left\{x|\left(x\in A\wedge \neg x\in A\right)\right\}$$
7	4 5 6	3eqtr4i	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$