Theorem dfnul4

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 3-Sep-2024)

	Ref	Expression
Assertion	dfnul4	⊢ ∅ = { 𝑥 ∣ ⊥ }

Proof

Step	Hyp	Ref	Expression
1		dfnul2	⊢ ∅ = { 𝑥 ∣ ¬ 𝑥 = 𝑥 }
2		equid	⊢ 𝑥 = 𝑥
3	2	notnoti	⊢ ¬ ¬ 𝑥 = 𝑥
4	3	bifal	⊢ ( ¬ 𝑥 = 𝑥 ↔ ⊥ )
5	4	abbii	⊢ { 𝑥 ∣ ¬ 𝑥 = 𝑥 } = { 𝑥 ∣ ⊥ }
6	1 5	eqtri	⊢ ∅ = { 𝑥 ∣ ⊥ }