We define $D( B(H))=\cup_n P(M_n( B(H)))/\sim$, where $ P(M_n(B(H))$ is the set of projections in $M_n(B(H))$,$\sim$ is the equivalence relation as follows:suppose $p$ is a projection in $P(M_n(B(H)))$,$q$ is a projection in $P(M_m(B(H)))$.$p\sim q$ if there is an element $v$ in $M_{m,n}(B(H))$ with $p=v^*v,q=vv^*$,$H$ is an infinite dimensional inseparable Hilbert space.
How to show that $D(B(H))$ is isomorphic to the semigroup of all cardinal numberss less than or equal to $dim(H)$