stuck in Foundation of Mathematics course I'm studying Foundation of Mathematics on my own due to personal situation and can't go to class



In the Reference book that I'm studying there is one portion of problem solution I can't understand.
here is the problem. 
$$\{x^2 -3 | x \in [-5, 1)\} = [-3, 22]$$

I understand whole proof and it's not hard at all except the part that:
$$-5 \leq x \lt 1 \space \text{ then } -5 \leq x \leq 5$$
I don't know why ?
$1\lt 5$, so if $x&lt;1$ then surely $x&lt;5,$ which implies $x\le 5$
Let $A= \{ x| -5 \le x &lt; 1 \}$ and $B = \{ x|-5 \le x \le 5 \}$.

$$A \subset B$$

$$x \in A \Rightarrow x \in B.$$