Let $A$ be a $C^*$-algebra. If necessary, let us assume that $A$ is a von Neumann algebra. For projections $p,q \in A$ one writes $p \prec q$ if $p$ is Murray-von Neumann-equivalent to a subprojection of $q$, i.e. there is a partial isometry $u \in A$ such that $u^* u = p$ and $u u^* \leq q$. One also says that $p$ is subordinate to $q$.
Why is $\prec$ a transitive relation?
Suppose $p\prec q\prec r$. That is, there exist projections $q_0\leq q$ and $r_0\leq r$ with partial isometries $u,v$ with $u^*u=p$, $uu^*=q_0$, $v^*v=q$, $vv^*=r_0$.
Note that $u=up=q_0u$ and $v=r_0v=vq$.
Let $w=vu$. Then $$ w^*w=u^*v^*vu=u^*qu=u^*qq_0u=u^*q_0u=u^*u=p, $$ and $$ ww^*=vuu^*v^*=vq_0v^*\leq vv^*=r_0\leq r. $$ So $p\prec r$.