Theorem ab0

Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne ). (Contributed by BJ, 19-Mar-2021)

	Ref	Expression
Assertion	ab0	⊢ ( { 𝑥 ∣ 𝜑 } = ∅ ↔ ∀ 𝑥 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜑 }
2	1	eq0f	⊢ ( { 𝑥 ∣ 𝜑 } = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ { 𝑥 ∣ 𝜑 } )
3		abid	⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
4	3	notbii	⊢ ( ¬ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ ¬ 𝜑 )
5	4	albii	⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ¬ 𝜑 )
6	2 5	bitri	⊢ ( { 𝑥 ∣ 𝜑 } = ∅ ↔ ∀ 𝑥 ¬ 𝜑 )