I think all proofs are just "convincing arguments." Computation would make that seem meaningless, but computers don't operate based on truth, they are repetition machines, just like you and I. But I've been reading an Introductory Topology book recently (Munkres) and the entire first chapter is explicating at a relatively high level of detail all the things you describe and moreso. But its necessary, in some sense. A lot of the set theoretical rules were employed in topology to create more generalized forms of analysis. You have to learn them because otherwise its very difficult to understand the language that's being used. But as I've argued elsewhere, if we are aware that its all based on interpretation, then I think we need to be a bit anarchistic and let students have the freedom to break those rules without knowing it, if they might produce thereby fascinating proofs and arguments. If you give people a schematic for interpretation, they're just going to follow it, and they might have difficulty deviating from it at all. The spark of human creativity comes when people are forced to, as they say, re-derive certain theorems, and while doing so they might accidentally discover some new implication of the logic that was unseen before, and produce something entirely new.