ThmDex – An index of mathematical definitions, results, and conjectures.
Result R47 on D70: Set difference
Set difference is subset of the minuend
Formulation 0
Let $X$ and $Y$ each be a D11: Set.
Then \begin{equation} X \setminus Y \subseteq X \end{equation}
Proofs
Proof 0
Let $X$ and $Y$ each be a D11: Set.
If $X \setminus Y$ is empty, then it is a subset of $X$ due to R7: Empty set is subset of every set. If $X \setminus Y$ is not empty, then $x$ being an element of $X \setminus Y$ implies that $x$ is an element of $X$ by the definition of set difference. Since this is true for every $x \in X \setminus Y$, the claim follows. $\square$