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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	⊢ ¬ 𝜑
	Assertion	abf	⊢ { 𝑥 ∣ 𝜑 } = ∅

Proof

Step	Hyp	Ref	Expression
1		abf.1	⊢ ¬ 𝜑
2	1	bifal	⊢ ( 𝜑 ↔ ⊥ )
3	2	abbii	⊢ { 𝑥 ∣ 𝜑 } = { 𝑥 ∣ ⊥ }
4		dfnul4	⊢ ∅ = { 𝑥 ∣ ⊥ }
5	3 4	eqtr4i	⊢ { 𝑥 ∣ 𝜑 } = ∅