You're missing something by reading this in an insufficiently pedantic way.

The pedantic way is as follows. The theorem's claim is "If A, then B", where A is the hypothesis. Claiming that A is false does not invalidate the theorem; in fact, if A could be proven to be false, then "If A, then B" would be vacuously true, and so in a way, arguing with the hypotheses only supports the theorem.

You could, however, claim that the theorem is useless if the hypothesis never holds. One example of this is "If 2+2=5, then the moon is made of green cheese". This is a true statement, but it doesn't tell us anything about the moon because 2+2 is not 5.