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) Prove directly from definition to allow shortening dfnul2 . (Revised by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul4	⊢ ∅ = { 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	3	bifal	⊢ ( ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V ) ↔ ⊥ )
5	4	abbii	⊢ { 𝑥 ∣ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ V ) } = { 𝑥 ∣ ⊥ }
6	1 2 5	3eqtri	⊢ ∅ = { 𝑥 ∣ ⊥ }