Theorem elneq

Description: A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022)

	Ref	Expression
Assertion	elneq	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elirr	⊢ ¬ 𝐵 ∈ 𝐵
2		nelelne	⊢ ( ¬ 𝐵 ∈ 𝐵 → ( 𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵 )