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)

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

Proof

Step	Hyp	Ref	Expression
1		df-nul	⊢ ∅ = ( V ∖ V )
2		df-dif	⊢ ( V ∖ V ) = { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V ) }
3		pm3.24	⊢ ¬ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V )
4		equid	⊢ 𝑥 = 𝑥
5	4	notnoti	⊢ ¬ ¬ 𝑥 = 𝑥
6	3 5	2false	⊢ ( ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V ) ↔ ¬ 𝑥 = 𝑥 )
7	6	abbii	⊢ { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V ) } = { 𝑥 ∣ ¬ 𝑥 = 𝑥 }
8	1 2 7	3eqtri	⊢ ∅ = { 𝑥 ∣ ¬ 𝑥 = 𝑥 }