I was trying to understand Curry's Paradox on wikipedia.
The example given is:
To produce Curry's paradox, as described in the two steps above, apply this method to the sentence "if this sentence is true, then Germany borders China". Here A, "this sentence is true", refers to the overall sentence, while B is "Germany borders China". So, assuming A is the same as assuming "If A, then B". Therefore, in assuming A, we have assumed both A and "If A, then B". Therefore B is true, by modus ponens, and we have proven "If this sentence is true, then 'Germany borders China' is true." in the usual way, by assuming the hypothesis and deriving the conclusion.
Now, because we have proved "If this sentence is true, then 'Germany borders China' is true", then we can again apply modus ponens, because we know that the claim "this sentence is true" is correct. In this way, we can deduce that Germany borders China.
I went through the formal proof as well, but what I fail to understand is that we still haven't proved that the sentence A is True, just assumed it to be True.
Since the resulting statements give us a paradox, it follows that the statement A can't be True in the first place. Where am I getting this wrong?