Let $M \subseteq \mathcal{L}(H)$ be a von Neumann algebra, and let $p \in M$ be a projection. The following are equivalent:
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1. $p$ is finite in $M$.
2. For all projections $e,f \in M$ with $e \leq f \leq p$, if $e \sim f$ in $M$, then $e=f$.
3. The projection $p$ is finite as the identity projection of the corner von Neumann algebra $pMp$.