Theorem eq0

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 NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

	Ref	Expression
Assertion	eq0	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dfnul4	⊢ ∅ = { 𝑦 ∣ ⊥ }
2	1	eqeq2i	⊢ ( 𝐴 = ∅ ↔ 𝐴 = { 𝑦 ∣ ⊥ } )
3		biidd	⊢ ( 𝑦 = 𝑥 → ( ⊥ ↔ ⊥ ) )
4	3	eqabbw	⊢ ( 𝐴 = { 𝑦 ∣ ⊥ } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ⊥ ) )
5		nbfal	⊢ ( ¬ 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ↔ ⊥ ) )
6	5	albii	⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ⊥ ) )
7	4 6	bitr4i	⊢ ( 𝐴 = { 𝑦 ∣ ⊥ } ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
8	2 7	bitri	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )