Is $A \subseteq C \land B \subseteq C \to A\cup B \subseteq C$ valid?

Question

My proof is $C \cup A = C \cup B = C$

$(x \in A\cup B \to x \in C)$ $\Rightarrow (x \in A\cup B\cup C \to x \in C)$ $\Rightarrow (x \in C \to x \in C)$

so $(x \in A\cup B \to x \in C)$ is true.

Is that valid? I'm not sure about my solution.

Answer 1

Your proof looks fine, perhaps add $A\cup B\cup C=(A\cup C) \cup (B\cup C)$

I would write it as:

$x \in A\cup B \implies x \in A\lor x\in B \implies x \in C\lor x\in C\implies x \in C$

$\therefore A\cup B\subseteq C$

Answer 2

You want to prove $x\in A\cup B\to x\in C$, so that expression shouldn't be your premise, nor should you adopt notation that gives the impression it is. The clearest way to explain why the theorem is true is to subdivide $x\in A\cup B$ into two cases. The first, $x\in A$, implies $x\in C$ because $A\subseteq C$. The second, $x\in B$, works similarly.