Does line belongs to plane?

I have to prove this:

If p and q are two parallel lines, line p doesn't intersect plane alpha, and line q and plane alpha have at least one common point, prove that line q belongs to plane alpha.

My proof:

Lines p and q determine one plane beta.

Then planes alpha and beta have one common point A (A belongs to line q).

Then there is common line q' for alpha and beta.

Line p is parallel to line q' (otherwise it would intersect plane alpha) but then is q'=q beacuse only one line is parallel to line p which goes throught point A, so q belongs to plane alpha.

Is my proof correct?