# Check if $A \subseteq B \land C \subseteq D \Rightarrow A \cap C \subseteq B \cap D$

Check if $A \subseteq B \land C \subseteq D \Rightarrow A \cap C \subseteq B \cap D$

My solution:
Consider an arbitrary object $x$. Assume that $x$ is in $A$ and that $x$ is in $C$. Then, by Modus Ponens, we can deduce that $x$ is in $B$ and in $D$. And so the fact that $x$ is in $A$ and $C$ simultaneously entails the fact that this object must be in $B$ and $C$ at the same time as well. The fact that $x$ is in $A$ and $C$, by definition, means that $x \in A\cap C$ and by deduction we got that $x \in B \cap D$
$\square$

Is my solution correct?

• Looks good to me. I would, for clarity, start by stating that $x\in A\cap C$. Then from there deduce that $x\in A$ and $x\in C$. Just to make it clear that you're trying to prove $x\in A\cap C\implies x\in B\cap D$, because it took me a couple of seconds to see that that's what you're actually doing and it shouldn't have. The way you've written it, it seems like you're beginning with $x\in A\land x\in C$, and that's not the exact starting point that you want. – Arthur Feb 1 '18 at 11:26
• Aemilius. Suggestion: You can(do not have to) make your approach clearer by using symbols like Arthur,and showing the different steps in new lines.if you read a page it helps if subdivided by paragraphs:)) – Peter Szilas Feb 1 '18 at 12:20