Completeness theorem states that consistent countable FO theory has a model. Compactness theorem states that FO theory has a model iff every finite subset of FO theory has a model. Both theorems are provable in ZFC.
Therefore:
Consistent(ZFC) <-> all finite subsets of ZFC have a model ->
not Consistent(ZFC) <-> some finite subsets of ZFC don’t have a model ->
some finite subsets of ZFC + not Consistent(ZFC) don’t have a model <->
Completeness theorem states that consistent countable FO theory has a model. Compactness theorem states that FO theory has a model iff every finite subset of FO theory has a model. Both theorems are provable in ZFC.
Therefore:
Consistent(ZFC) <-> all finite subsets of ZFC have a model ->
not Consistent(ZFC) <-> some finite subsets of ZFC don’t have a model ->
some finite subsets of ZFC + not Consistent(ZFC) don’t have a model <->
not Consistent(ZFC + not Consistent(ZFC)),
proven in ZFC + not Consistent(ZFC)