Modern hardware designs have grown increasingly efficient and complex. However, they are often susceptible to Common Weakness Enumerations (CWEs). This paper is focused on the formal verification of CWEs in a dataset of hardware designs written in SystemVerilog from Regenerative Artificial Intelligence (AI) powered by Large Language Models (LLMs). We applied formal verification to categorize each hardware design as vulnerable or CWE-free. This dataset was generated by 4 different LLMs and features a unique set of designs for each of the 10 CWEs we target in our paper. We have associated the identified vulnerabilities with CWE numbers for a dataset of 60,000 generated SystemVerilog Register Transfer Level (RTL) code. It was also found that most LLMs are not aware of any hardware CWEs; hence they are usually not considered when generating the hardware code. Our study reveals that approximately 60% of the hardware designs generated by LLMs are prone to CWEs, posing potential safety and security risks. The dataset could be ideal for training LLMs and ML algorithms to abstain from generating CWE-prone hardware designs.