Theorem nssne2

Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015)

	Ref	Expression
Assertion	nssne2	$$⊢\left(A\subseteq C\wedge \neg B\subseteq C\right)\to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		sseq1	$$⊢A=B\to \left(A\subseteq C\leftrightarrow B\subseteq C\right)$$
2	1	biimpcd	$$⊢A\subseteq C\to \left(A=B\to B\subseteq C\right)$$
3	2	necon3bd	$$⊢A\subseteq C\to \left(\neg B\subseteq C\to A\ne B\right)$$
4	3	imp	$$⊢\left(A\subseteq C\wedge \neg B\subseteq C\right)\to A\ne B$$