ZFC+¬Con(ZFC) proves "ZFC+¬Con(ZFC) is inconsistent". So if were the case that ZFC+¬Con(ZFC) was sound then what it proves would be true, and "ZFC+¬Con(ZFC) is

ZFC+¬Con(ZFC) proves "ZFC+¬Con(ZFC) is inconsistent". So if were the case that ZFC+¬Con(ZFC) was sound then what it proves would be true, and "ZFC+¬Con(ZFC) is inconsistent" would be true. But that would mean ZFC+¬Con(ZFC) is actually inconsistent, and thus ZFC+¬Con(ZFC) would actually be unsound after all.