If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is added to it as a new axiom?

"There exists a denumerably infinite and ordinal definable set of real numbers, not all of whose elements are ordinal definable"

If the answer to the above question is negative, then it must be provable in ZFC that every denumerably infinite and ordinal definable set of real numbers is hereditarily ordinal definable. This is because every real number can be regarded as a set of finite ordinal numbers and every finite ordinal number is ordinal definable. Garabed Gulbenkian