Calculations for measuring a two-qubit system

Suppose I have the state $|psirangle = frac{1}{sqrt{2}}(|01rangle – |10rangle)$ that I want to measure in an arbitrary basis $$|Arangle = alpha|0rangle + beta|1rangle text{ and } |Brangle = beta^*|0rangle – alpha^*|1rangle$$

From my understanding, if I measure $|psirangle$, the probability of seeing $|Arangle$ is $$langle A | psirangle^2$$ But when I try to compute $langle A | psirangle$, I get

begin{align*}
langle A | psirangle &= frac{1}{sqrt{2}}(alpha^*langle 0 | + beta^* langle 1 |)(|01rangle – |10rangle)
&= frac{1}{sqrt{2}}(alpha^*langle 0 | + beta^* langle 1 |)(|01rangle – |10rangle)
&= frac{1}{sqrt{2}}(alpha^*langle 0 | + beta^* langle 1 |)(|0rangleotimes|1rangle – |1rangleotimes|0rangle)
end{align*}

I assume I can distribute, so I get terms like $alpha^*langle 0 |big(|0rangleotimes|1ranglebig)$.

But how does one take an inner product between $|0rangle$ and $|0rangleotimes|1rangle$, when the latter of the two is an element of a tensor product space of different dimension as $|0rangle$?