Valid: The number is at least 5 or at most 5; it is not at least 5; therefore it is at most 5.
Invalid: The number is at least 5 or at most 5; it is at least 5; therefore it is not at most 5.
The number 5 is a counterexample: it is both at least 5 and at most 5.
What this kind of disjunction guarantees is that both cannot be denied. That is why denial of one yields affirmation of the other.
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Exceptive Syllogism: Valid Inferences from Connected and Disjunctive Conditionals
Valid: The number is at least 5 or at most 5; it is not at least 5; therefore it is at most 5.
Invalid: The number is at least 5 or at most 5; it is at least 5; therefore it is not at most 5.
The number 5 is a counterexample: it is both at least 5 and at most 5.
What this kind of disjunction guarantees is that both cannot be denied. That is why denial of one yields affirmation of the other.