Theorem eq0f

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by BJ, 15-Jul-2021)

	Ref	Expression
Hypothesis	eq0f.1	⊢ Ⅎ 𝑥 𝐴
Assertion	eq0f	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eq0f.1	⊢ Ⅎ 𝑥 𝐴
2		nfcv	⊢ Ⅎ 𝑥 ∅
3	1 2	cleqf	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
4		noel	⊢ ¬ 𝑥 ∈ ∅
5	4	nbn	⊢ ( ¬ 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
6	5	albii	⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅ ) )
7	3 6	bitr4i	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )