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The deployment of 5G has exposed capacity constraints in realizing the key vision of the Internet of Everything (IoE). Therefore, the researchers are exploring potentials of Digital Twin Network (DTN) in wireless networks. DTN is a novel technology to create virtual replicas of physical environment for testing, optimizing, and managing wireless networks. The integration of Artificial Intelligence (AI) and DTN appears to be a promising approach to address communication systems by providing an efficient environment for testing and improving AI models before deployment in real networks for effective network management, optimal resource allocation, and precise behavior prediction. Therefore, AI-enabled DTN in 6G represents a compelling avenue to address multifaceted challenges faced by wireless networks. In this comprehensive work, we offer a holistic survey that delves into the state-of-the-art approaches for AI-enabled DTNs in 6G. Firstly, we discuss the evolution of wireless networks and concept of AI-enabled DTN in 6G. Secondly, we discuss the role of AI-enabled DTN in 6G and driving advancements in fundamental components of 6G including resource allocation, caching, data offloading, and data security. Thirdly, we conduct a detailed discussion on key enabling technologies for realizing the capabilities of AI-enabled DTN in 6G. Fourthly, several applications of AI-enabled DTN in 6G are discussed for the practical relevance and significance in various industries such as smart cities, healthcare, and transportation etc. Finally, we provide lessons learned and highlight existing challenges and research directions to embark on further research efforts in the realm of AI-enabled DTN in 6G.