Theorem elnelneqd

Description: Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypotheses	elnelneqd.1	$$⊢\phi \to C\in A$$
	elnelneqd.2	$$⊢\phi \to \neg C\in B$$
Assertion	elnelneqd	$$⊢\phi \to \neg A=B$$

Proof

Step	Hyp	Ref	Expression
1		elnelneqd.1	$$⊢\phi \to C\in A$$
2		elnelneqd.2	$$⊢\phi \to \neg C\in B$$
3	1	adantr	$$⊢\left(\phi \wedge A=B\right)\to C\in A$$
4		simpr	$$⊢\left(\phi \wedge A=B\right)\to A=B$$
5	3 4	eleqtrd	$$⊢\left(\phi \wedge A=B\right)\to C\in B$$
6	2 5	mtand	$$⊢\phi \to \neg A=B$$