TITLE: Can ZFC+A and ZFC+negation of A be both inconsistent where A is some conjecture?
QUESTION [5 upvotes]: So I know that a conjecture/statement or negation of it plus ZFC can both turn out to be consistent, which means that a statement is not provable. 
But I would like to go opposite way - and let's say they are both inconsistent. First of all, is this possible, and secondly, if there is an example, what would be a such case?

REPLY [1 votes]: If a theory proves $X$ and it proves $Y$, it proves $X \land Y$ (this is how $X \land Y$ is typically proven). So if it proves $A$ and $\lnot A$, it proves $A \land \lnot A$, meaning ZFC would be inconsistent.
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