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 GG, 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	1	bifal	$$⊢\phi \leftrightarrow \perp$$
3	2	abbii	$$⊢\left\{x|\phi \right\}=\left\{x|\perp \right\}$$
4		dfnul4	$$⊢\varnothing =\left\{x|\perp \right\}$$
5	3 4	eqtr4i	$$⊢\left\{x|\phi \right\}=\varnothing$$