Theorem dfnul3OLD

Description: Obsolete version of dfnul4 as of 23-Sep-2024. (Contributed by NM, 25-Mar-2004) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	dfnul3OLD	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$

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	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$