They make doggy doors that come with collars and only the pets with the collars can go out the door. It is to keep strange animals out of the house basically but would work with this too. The cat could not use the door and the dogs could. Good luck and let us know.
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PHWildCat/Tessa and 22 cats, 1 dog and even a hamster
Tabitha, in my heart forever.... earned her wings 6/26/05, Samantha, the bestest whineameezer...earned her wings 4/7/06. Penny the little calico earned her wings 12/12/07 and Patch of Beauty Onyxcita the Wimp (Onyx) who flew to the bridge on 11/5/08 at the age of 21 years.
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