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Metamath Proof Explorer

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	$⊢ \neg φ$
	Assertion	abf	$⊢ \{x \| φ\} = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		abf.1	$⊢ \neg φ$
2	1	bifal	$⊢ φ \leftrightarrow ⊥$
3	2	abbii	$⊢ \{x \| φ\} = \{x \| ⊥\}$
4		dfnul4	$⊢ \emptyset = \{x \| ⊥\}$
5	3 4	eqtr4i	$⊢ \{x \| φ\} = \emptyset$