[proofplan] We use the defining projection-theoretic characterization of a Type III factor: a factor is Type III exactly when it has no nonzero finite projections. This immediately identifies Type III with the assertion that every nonzero projection is infinite. The remaining point is that a nonzero corner $pMp$ is finite exactly when its identity projection $p$ is finite, and this finiteness is equivalent to finiteness of $p$ as a projection of $M$. [/proofplan]

[step:Translate Type III into absence of nonzero finite projections] By definition, a factor $M$ is Type III if and only if $M$ has no nonzero finite projections. A projection is infinite precisely when it is not finite. Therefore the statement that $M$ is Type III is equivalent to the statement that every nonzero projection $p\in M$ is infinite. [/step]

[step:Identify finite corners with finite projections]Let $p\in M$ be a nonzero projection. The corner \begin{align*} pMp=\{pxp:x\in M\} \end{align*} is a von Neumann algebra with identity projection $p$. By the finite-projection corner criterion [citetheorem:9280], the projection $p$ is finite in $M$ if and only if $p$ is finite as the identity projection of the corner von Neumann algebra $pMp$. By the definition of a finite von Neumann algebra, $pMp$ is finite if and only if its identity projection $p$ is finite. Hence \begin{align*} p \text{ is finite in } M \iff pMp \text{ is a finite von Neumann algebra}. \end{align*}[/step]

[guided]Fix a nonzero projection $p\in M$. The corner algebra is \begin{align*} pMp=\{pxp:x\in M\}. \end{align*} Its multiplicative identity is $p$, because for every element $pxp\in pMp$ one has \begin{align*} p(pxp)=pxp \end{align*} and \begin{align*} (pxp)p=pxp. \end{align*} Now we compare two possible meanings of finiteness for $p$. First, $p$ may be finite as a projection in the ambient von Neumann algebra $M$. Second, $p$ may be finite as the identity projection of the smaller von Neumann algebra $pMp$. These are equivalent by the finite-projection corner criterion [citetheorem:9280], applied to the projection $p\in M$. Finally, by the convention for finite von Neumann algebras, a von Neumann algebra is finite exactly when its identity projection is finite. Since the identity of $pMp$ is $p$, this gives \begin{align*} pMp \text{ is finite } \iff p \text{ is finite as the identity of } pMp. \end{align*} Combining this with the corner criterion yields \begin{align*} pMp \text{ is finite } \iff p \text{ is finite in } M. \end{align*} This is the bridge between the projection condition and the corner condition.[/guided]

[step:Conclude that nonzero finite corners and nonzero finite projections are the same obstruction] Suppose first that every nonzero projection in $M$ is infinite. Let $p\in M$ be a nonzero projection. Then $p$ is not finite in $M$, so by the equivalence established above, $pMp$ is not a finite von Neumann algebra. Thus condition 2 implies condition 3. Conversely, suppose condition 3 holds. If $M$ had a nonzero finite projection $p\in M$, then the equivalence established above would imply that $pMp$ is a finite von Neumann algebra. This contradicts condition 3. Hence $M$ has no nonzero finite projections, so every nonzero projection in $M$ is infinite. Therefore condition 3 implies condition 2. Together with the first step, conditions 1, 2, and 3 are equivalent. [/step]