Here is a short demonstration.
Select balls until the first red or yellow ball is picked. (We assume this is certain to happen.) Regardless of how balls are selected (and you can even play Polya urn-like games and put more balls back into the urn after each selection), let us suppose that so long as no red or yellow balls are picked, there always remain $r$ red balls and $y$ yellow balls in the urn and that all red and yellow balls are equally likely to be chosen. (This is the case with independent random selection, either with or without replacement.) This assumption implies that at the moment a red or yellow ball is first selected, it has $r$ chances of being red out of the $r+y$ possible balls, giving a probability of $r/(r+y)$. Done!
You asked for a proof. OK, that's fair: rigor requires mathematical notation and concepts because they help us avoid mistakes. The following analysis, which emulates the preceding demonstration, is extremely general but requires only definitions and axioms and therefore is completely elementary.
Let $$X_0, X_1, X_2, \ldots, X_n, \ldots$$ be a stochastic process that models the selection of balls. For each $i$ let $\mathcal{R}_i \subset \mathbb{R}$ and $\mathcal{Y}_i \subset \mathbb{R}$ be disjoint Borel sets modeling the events "a red ball is drawn at step $i$" and "a yellow ball is drawn at step $i$," respectively. Let $\mathcal{B}_i = \mathbb{R}\setminus \left(\mathcal{R}_i \cup \mathcal{Y}_i\right)$ be the event in which neither a red ball nor a yellow ball is drawn at step $i$. The event "no red or yellow balls are drawn before step $i$ but one of them is drawn at step $i$" is
$$\mathcal{A}_{i}=\{\omega\,|\, X_0(\omega) \in \mathcal{B}_0, X_1(\omega)\in\mathcal{B}_1,\ldots,X_{i-1}(\omega)\in\mathcal{B}_{i-1}, X_{i}(\omega)\in\mathcal{R}_i \cup\mathcal{Y}_i\}.$$
The chance of drawing a red ball at step $i$, after seeing no red or yellow balls before then, can be written in terms of the conditional probability
$$\Pr(X_i \in \mathcal{R}_i\,|\, \mathcal{A}_{i}) \Pr(\mathcal{A}_{i}).\tag{1}$$
By definition, if $\mathcal{A}_i$ occurs, then either $\mathcal{R}_i$ or $\mathcal{Y}_i$ must occur--and these events are disjoint. Therefore
$$\Pr(\mathcal{R}_i\,|\,\mathcal{A}_i) = \frac{\Pr(\mathcal{R}_i)}{\Pr(\mathcal{R}_i) + \Pr(\mathcal{Y}_i)}.\tag{2}$$
Let us suppose it is certain that eventually a red or yellow ball is drawn. Because the $\mathcal{A}_i$ are disjoint, this can be expressed
$$1 = \sum_{i=0}^\infty \Pr(\mathcal{A}_i).\tag{3}$$
The chance of drawing a red ball first is obtained (according to the definition of conditional probability) by summing over all possible $\mathcal{A}_i$, because these are disjoint and exhaustive (apart perhaps from a set of probability zero representing the event no red or yellow ball is ever drawn). Combining $(1)$ and $(2)$ gives the expression
$$\Pr(\text{Red drawn first}) = \sum_{i=0}^\infty \frac{\Pr(\mathcal{R}_i)}{\Pr(\mathcal{R}_i) + \Pr(\mathcal{Y}_i)} \Pr(\mathcal{A}_i).\tag{4}$$
This is the general solution we were aiming for.
When sampling randomly, with or without replacement, from an urn with $r$ red balls and $y$ yellow balls, suppose that at step $i$ there are $b_i \ge 0$ additional balls of any (non-red, non-yellow) color. Then
$$\Pr(\mathcal{R}_i) = \frac{r}{r + y + b_i}$$
and
$$\Pr(\mathcal{Y}_i) = \frac{y}{r + y + b_i},$$
whence
$$\frac{\Pr(\mathcal{R}_i)}{\Pr(\mathcal{R}_i) + \Pr(\mathcal{Y}_i)} = \frac{r/(r+y+b_i)}{r/(r+y+b_i) + y/(r+y+b_i)} = \frac{r}{r+y}.$$
Consequently $(4)$, by virtue of $(3)$, simplifies to
$$\Pr(\text{Red drawn first}) = \sum_{i=0}^\infty\frac{r}{r+y}\Pr(\mathcal{A}_i)=\frac{r}{r+y}\sum_{i=0}^\infty\Pr(\mathcal{A}_i) = \frac{r}{r+y}.$$
That finishes the proof when sampling from a finite urn without replacement, for it is certain that eventually a red or yellow ball will be drawn (assuming there is at least one of them in the urn). When sampling with replacement, a standard argument (based on the geometric distribution, but of only tangential interest here) shows that eventually drawing a red or yellow ball is certain provided the chances are not zero to begin with. That will be the case when sampling from any finite urn, QED.
