We derive a full Ising version of the t-J model in which the hopping t-term possesses the global discrete Z_2 symmetry. This results in a more tractable though still nontrivial model that captures the physic of strong correlations. This model can be thought of as a doped Ising model. It allows for the unbiased Monte Carlo calculations on large clusters of up to 10^3 sites. Within this model, we discuss in detail the destruction of the antiferromagnetic (AF) order by doping as well as the interplay between the AF order and hole mobility.  We explicitly demonstrate that strong electron correlations play a key role in destroying the AF order at hole concentration below 6%, which remains in a perfect  agreement with experimental data on the underdoped cuprates.