I heard this story and found it interesting. A tenant got a dog. The landlord found out and gave the tenants 10 days to move out. Is this legal? From the relevant legislation it says it is possible for leases to expresily prohibit pets. Assuming the lease does not expresiley prohibit or allow pets, can a landlord terminate tenancy because the tenant has a pet? If yes, what is the amount of time required to allow the tenants to move out? If the tenant remedies the issue (e.g. finds another home for the pet) can the landlord still terminate tenancy?
From the page you linked, if a lease doesn't prohibit pets then pets are allowed unless they cause damage or other problems. If pets are allowed and they are not causing damage or problems the landlord can take no action.
The remainder of your question is answered by following the link from your link about getting a pet without permission:
If the tenancy agreement says that pets aren’t allowed and the tenant gets a pet, two things could happen:
The landlord may give the tenant a “breach letter” that explains how the agreement has been broken, how much time is allowed to remove the pet and what will happen if the pet is not removed (e.g. eviction)
The landlord and tenant may agree to change this term and record it in the agreement