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"The exception that proves the rule" is an exception to a generally accepted truth. This is an old fashioned use of the word 'prove', which means 'to test'. It does not mean that it demonstrates a rule to be true, but that it tests the rule. It is usually used these days when an exception to a rule has been identified: for example, Mutillidae are wasps without wings, and therefore are an exception that proves (tests) the rule that wasps fly.
"The exception [that] proves the rule" also means that the presence of an exception applying to a specific case establishes ("proves") that a general rule exists. For example, a sign that says "parking prohibited on Sundays" (the exception) "proves" that parking is allowed on the other six days of the week (the rule). A more explicit phrasing might be "the exception that proves the existence of the rule."