When a land lords sells to an owner occupiers does it push up rents? In the article Landlords who own one property ‘left without relief’ under new Victorian Government package posted on 15 April 2020 on the domain website. Barry Plant chief executive Mike McCarthy Makes the claim that landlords selling to owner-occupiers will push up the price of rent by reducing the available pool of rental properties as below. 


  “We are going to have landlords that are going to have to sell their
  property and while that’s great for first-home buyers what it does is
  reduce the pool of properties for tenants and over time that will push
  up rents,” Mr Webb said.


There is a similar claim which was made by PETER VERWER chief executive of the Australian property council in this 2013 discussion about a Grattan institute report (Check the transcript for full details. 


  They create 1.8 million homes available for rent. They’re not new homes, but they are 1.8million homes that would not exist unless a
  private investor is satisfied with the high compliance costs, the
  quite moderate rent and the maintenance costs to create all of this
  rented space


Does this claim add up, my understanding is that its a zero sum game. e.g. A landlord sells the property and a owner occupier moves in. Then there is one less property available for rent however one less person (or group of people) looking for somewhere to rent. 

OR am I missing something? 

Notes this is based on content in Australia. 

https://www.domain.com.au/news/landlords-left-without-relief-under-new-victorian-government-package-949032/
Your argument is sound (as a first approximation). In fact this is one of the basic examples discussed in chapter 1 of Varian's Intermediate Microeconomics (a standard textbook). 


  Suppose that a developer decides to turn several of the apartments
  into condominiums. What will happen to the price of the remaining
  apartments? Your first guess is probably that the price of apartments
  will go up, since the supply has been reduced. But this isn’t
  necessarily right. It is true that the supply of apartments to rent
  has been reduced. But the demand for apartments has been reduced as
  well, since some of the people who were renting apartments may decide
  to purchase the new condominiums.


(quote from pg 10 of the 8th edition) 

If you make some additional assumptions e.g., that all buyers are former tenants and that there is no friction in the market (rent changes may be larger when there are new tenants to avoid tension), the equilibrium price will be the same.