Theorem abf

Description: A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012) Avoid ax-8 , ax-10 , ax-11 , ax-12 . (Revised by Gino Giotto, 30-Jun-2024)

	Ref	Expression
Hypothesis	abf.1	$$⊢\neg \phi$$
Assertion	abf	$$⊢\left\{x|\phi \right\}=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		abf.1	$$⊢\neg \phi$$
2		equid	$$⊢x=x$$
3	2	notnoti	$$⊢\neg \neg x=x$$
4	1 3	2false	$$⊢\phi \leftrightarrow \neg x=x$$
5	4	abbii	$$⊢\left\{x|\phi \right\}=\left\{x|\neg x=x\right\}$$
6		dfnul2	$$⊢\varnothing =\left\{x|\neg x=x\right\}$$
7	5 6	eqtr4i	$$⊢\left\{x|\phi \right\}=\varnothing$$