1. The problem is to understand the axioms of ZFC (Zermelo-Fraenkel set theory with Choice) combined with the large cardinal axiom I0, which states the existence of a nontrivial elementary embedding $j: V_{\lambda+1} \to V_{\lambda+1}$ for some limit ordinal $\lambda$.\n\n2. ZFC axioms include Extensionality, Foundation, Pairing, Union, Power Set, Infinity, Replacement, and Choice. These axioms form the standard foundation of modern set theory.\n\n3. The I0 axiom is a very strong large cardinal axiom asserting the existence of an elementary embedding $j$ from $V_{\lambda+1}$ into itself with critical point less than $\lambda$. This implies the existence of huge large cardinals and has deep consequences in set theory.\n\n4. To "prove" ZFC + I0 is consistent or to analyze its consequences requires advanced techniques in inner model theory and large cardinal theory, which are beyond elementary proofs. The axiom I0 is independent of ZFC and cannot be proven or disproven within ZFC alone.\n\n5. Therefore, the statement "Prove ZFC + I0" is not a standard problem with a proof but rather a research-level assertion about the consistency and implications of adding I0 to ZFC.\n\n6. In summary, ZFC + I0 is a strong set-theoretic hypothesis extending ZFC with a large cardinal axiom that cannot be proven from ZFC alone. It is studied for its profound implications in the foundations of mathematics.