$x+1\neq x$ just follows from $\phi (x)$ but the former one is a problem. You will need a double induction to show that $x+y\not\in x$.

$x+1\neq x$ just follows from $\phi (x)$. Now you need $x+1\not\in x$. Suppose the opposite. Then $x+1\in x$ and by ordinality $x+1\subseteq x$. But this will imply that $x\in x$, contradiction with the induction assumption.