This is the full statement you say is wrong. How so? The class of ordinals within a model of ZFC would imply that $\omega_1$ cannot exist because either statement #1 is true or statement #2 is true and both statements assert that $\omega_1$ cannot exist. Statement #1 asserts that $\omega_1$ cannot exist and is independent of the axiom of choice. Statement #2 asserts that $\omega_1$ might exist given the negation of the axiom of choice, but that $\omega_1$ cannot exist given the axiom of choice.Assume statement #2 is true. Then we have an uncountable set, $\omega_1$, being equal to a countable union of countable sets. Assume the axiom of choice and statement #2 implies that $\omega_1$ is countable, contradicting the definition of $\omega_1$.