Which is the proper church?

The Pope has said that non-Catholics are not the proper church. Well, he didn’t actually say it, but a document with his name on does say it so for the sake of this post we’ll assume he did say it in the planning meetings for the document. I have been trying to get my head around this as it could now be the case that us non-Catholics have not been the proper church all along when we thought we were. It seems to me that our status as the proper church or otherwise depends largely upon whether the Pope was speaking infallibly when he said it. If I understand things correctly the Pope sometimes speaks infallibly and sometimes doesn’t.

Here is my analysis:

If the Pope was speaking infallibly and was right then the Catholics are the proper church, making us non-Catholics not the proper church.

If the Pope was speaking infallibly and was wrong, then he was not speaking infallibly. Therefore the Catholics are not the proper church. Us non-Catholics would therefore be the proper church, but the Catholics certainly would not be.

If the Pope was speaking fallibly and was right then it would at first appear that us non-Catholics are not the proper church. But, a later Pope could overturn this when speaking infallibly, in which case us non-Catholics would be the proper church even though we had thought that we were not the proper church. Of course this later Pope might be wrong, in which case the Catholics would be proved to be not the proper church, making us the proper church.

If the Pope was speaking fallibly and was wrong then us non-Catholics are the proper church. The Catholics could still be the proper church too though, as you can’t blame the Pope for getting it wrong when speaking fallibly. After all, we all do that.

Of course, it could still be the case in several of these instances that neither the Catholics or us non-Catholics are the proper church, but that a third party that we have not yet thought of are the proper church and neither us nor the Catholics are the proper church. I do not intend to discuss this though as it makes the situation unnecessarily complicated which I do not want to do.

[Minor update to remove mistake made due to fallibility]