You're on the right track! There's just one thing you're missing.
The "build all the strings and union them together" approach works great—if you have a finite number of strings. There's a theorem that says "a union of finitely many regular languages is regular", but the key is the finitely many.
In this case, there are infinitely many strings in the language, so the union-them-all trick no longer works.
If you want to prove that the language is not regular (as opposed to just failing to prove that it is regular), try a fooling set proof:
- Let $F$ be the language $1^*$. It's clearly infinite.
- Let $x$ and $y$ be two distinct strings in $F$. These can be written as $1^i$ and $1^j$ with $i \neq j$.
- Now, let $z$ be $0^i$. From the definition of your language, we can see that $xz$ is in the language, but $yz$ is not.
- Therefore, all the different strings in $F$ are distinguishable as prefixes. Since the language has infinitely many distinguishable prefixes, it cannot be regular.