Theorem n0el3

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021)

	Ref	Expression
Assertion	n0el3	⊢ ( ¬ ∅ ∈ 𝐴 ↔ ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		n0elim	⊢ ( ¬ ∅ ∈ 𝐴 → ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 )
2		n0eldmqseq	⊢ ( ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 → ¬ ∅ ∈ 𝐴 )
3	1 2	impbii	⊢ ( ¬ ∅ ∈ 𝐴 ↔ ( dom ( ◡ E ↾ 𝐴 ) / ( ◡ E ↾ 𝐴 ) ) = 𝐴 )