Magnon-photon coupling inside a cavity has been experimentally realized and has attracted significant attention for its potential docking with quantum information science. Whether this coupling implies the steady entanglement of photons and magnons is crucial for its usage in quantum information but is still an open question. Here we study the entanglement properties among magnons and photons in an antiferromagnet-light system and find that the entanglement between a magnon and a photon is nearly zero, while the magnon-magnon entanglement is very strong and can be even further enhanced through the coupling with the cavity photons. The maximum enhancement occurs when the antiferromagnet is resonant with the light. The essential physics can be well understood within the picture of cavity-induced cooling of the magnon-magnon state near its joint vacuum with stronger entanglement. Our findings can be used to cool magnetic magnons toward their ground state and may also be significant to extend the cavity spintronics to quantum manipulation. Furthermore, the hybrid antiferromagnet-light system provides a natural platform to manipulate the deep strong correlations of continuous modes with a generic stable condition and easy tunability.