If (A is a subset B) and (C is a subset of D), then (A intersection C) is a subset of (B intersection D).

I think I have a correct answer, but I'm uncertain (I feel like I'm leaving something out and that there should be something between the last two lines that explains further) so if you can answer/explain, that would be super helpful. Thanks in advance!

Here's what I have:

Let x be an element of A intersect C.

Since we know A is a subset of B, then we know that x is an element of A and therefore is an element of B too.

Also, since C is a subset of D and x is an element of C then x is also an element of D too.

This shows that A intersect C is a subset of B intersect D.