Theorem pssdifn0

Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994)

	Ref	Expression
Assertion	pssdifn0	$$⊢\left(A\subseteq B\wedge A\ne B\right)\to B\setminus A\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ssdif0	$$⊢B\subseteq A\leftrightarrow B\setminus A=\varnothing$$
2		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
3	2	simplbi2	$$⊢A\subseteq B\to \left(B\subseteq A\to A=B\right)$$
4	1 3	syl5bir	$$⊢A\subseteq B\to \left(B\setminus A=\varnothing \to A=B\right)$$
5	4	necon3d	$$⊢A\subseteq B\to \left(A\ne B\to B\setminus A\ne \varnothing \right)$$
6	5	imp	$$⊢\left(A\subseteq B\wedge A\ne B\right)\to B\setminus A\ne \varnothing$$