Theorem eqneltri

Description: If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021)

	Ref	Expression
Hypotheses	eqneltri.1	⊢ 𝐴 = 𝐵
	eqneltri.2	⊢ ¬ 𝐵 ∈ 𝐶
Assertion	eqneltri	⊢ ¬ 𝐴 ∈ 𝐶

Proof

Step	Hyp	Ref	Expression
1		eqneltri.1	⊢ 𝐴 = 𝐵
2		eqneltri.2	⊢ ¬ 𝐵 ∈ 𝐶
3	1	eleq1i	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 )
4	2 3	mtbir	⊢ ¬ 𝐴 ∈ 𝐶