Theorem dfnul2

Description: Alternate definition of the empty set. Definition 5.14 of TakeutiZaring p. 20. (Contributed by NM, 26-Dec-1996) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul2	⊢ ∅ = { 𝑥 ∣ ¬ 𝑥 = 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		dfnul4	⊢ ∅ = { 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	eqtr4i	⊢ ∅ = { 𝑥 ∣ ¬ 𝑥 = 𝑥 }