In this paper we generalise a useful result due to J. Mierczynski which states that for a strictly cooperative system on the positive orthant, with increasing first integral, all bounded orbits are convergent. Moreover there can be no more than one equilibrium on any level set, and any equilibrium attracts its entire level set. Here, more general state spaces and more general orderings are considered. Let Y be a subset of K which is a subset of R^n, where Y and K are proper cones. Given a local semiflow φ on Y which is strongly monotone with respect to K, and which preserves a K-increasing first integral, we show that every bounded orbit converges. Again, there can be no more than one equilibrium on any level set, and each equilibrium attracts its entire level set. An application from chemical dynamics is provided.