Theorem dfnul3

Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004)

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

Proof

Step	Hyp	Ref	Expression
1		pm3.24	$$⊢\neg \left(x\in A\wedge \neg x\in A\right)$$
2		equid	$$⊢x=x$$
3	1 2	2th	$$⊢\neg \left(x\in A\wedge \neg x\in A\right)\leftrightarrow x=x$$
4	3	con1bii	$$⊢\neg x=x\leftrightarrow \left(x\in A\wedge \neg x\in A\right)$$
5	4	abbii	$$⊢\left\{x|\neg x=x\right\}=\left\{x|\left(x\in A\wedge \neg x\in A\right)\right\}$$
6		dfnul2	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$
7		df-rab	$$⊢\left\{x\in A|\neg x\in A\right\}=\left\{x|\left(x\in A\wedge \neg x\in A\right)\right\}$$
8	5 6 7	3eqtr4i	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$