  We show that spin squeezing implies entanglement for quantum
tripartite-state, where the subsystem of the bipartite-state is identical. We
study the relation between spin squeezing parameters and entanglement through
the quantum entropy of a system starts initially in a pure state when the
cavity is binomial. We show that spin squeezing can be a convenient tool to
give some insight into the subsystems entanglement dynamics when the bipartite
subsystem interacts simultaneously with the cavity field subsystem, specially
when the interaction occurs off-resonantly without and with a nonlinear medium
contained in the cavity field subsystem. We illustrate that, in case of large
off-resonance interaction, spin squeezing clarifies the properties of
entanglement almost with full success. However, it is not a general rule when
the cavity is assumed to be filled with a non-linear medium. In this case, we
illustrate that the insight into entanglement dynamics becomes more clearly in
case of a weak nonlinear medium than in strong nonlinear medium. In parallel,
the role of the phase space distribution in quantifying entanglement is also
studied. The numerical results of Husimi $Q$-function show that the integer
strength of the nonlinear medium produces Schr\"{o}dinger cat states which is
necessary for quantum entanglement.
