An inequality about expectation

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Suppose $X, Y : \Omega \rightarrow [0, 1]$ are two random variables such that $\mathbb{E}(Y | X) = X$ and $\mathbb{E} X = 1 / 2$. Let $x, y \in (0, 1)$ be two real numbers such that
$$ \mathbb{E} [(1 - X) 1_{X \geq x}] = \mathbb{E} [(1 - Y) 1_{Y \geq y}]. $$ Can we show that $$ \mathbb{E} [X 1_{X \geq x}] \leq \mathbb{E} [Y 1_{Y \geq y}]? $$