Theorem difn0

Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017)

	Ref	Expression
Assertion	difn0	$$⊢A\setminus B\ne \varnothing \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		eqimss	$$⊢A=B\to A\subseteq B$$
2		ssdif0	$$⊢A\subseteq B\leftrightarrow A\setminus B=\varnothing$$
3	1 2	sylib	$$⊢A=B\to A\setminus B=\varnothing$$
4	3	necon3i	$$⊢A\setminus B\ne \varnothing \to A\ne B$$