Theorem elnelneq2d

Description: Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023)

	Ref	Expression
Hypotheses	elnelneq2d.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
	elnelneq2d.2	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 )
Assertion	elnelneq2d	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elnelneq2d.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2		elnelneq2d.2	⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 )
4	1	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝐶 )
5	3 4	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝐶 )
6	2 5	mtand	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )