Gödel's incompleteness theorem applys to ZFC theory

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When I assume ZFC's consistency, it is impossible to prove ZFC's consistency in itself from Gödel's incompleteness theorem 2. If ZFC's consistency have done, its proof need to be done in stronger system(ZFC's axioms can be proved in it) than ZFC. If this stronger system's consistency assumed to be consistency,its consistency can not be done in itself,be done in stronger System than it. Is it correct?