If $x\not\leq y$, then is $x>y$, or $x\geq y$? I'm currently reading about surreal numbers from here. 
At multiple points in this paper, the author has stated that if $x\not\leq y$, then $x\geq y$. 
Shoudn't the relation be "if $x\not\leq y$, then $x>y$"?
Hasn't the possibility of $x=y$ already been negated when we said $x\not\leq y$? 
Thanks in advance. 

Here's another way of looking at it.
$$(x\le{y})\leftrightarrow (x<y \lor x=y)$$
$$(x\not\le{y})\leftrightarrow(x>y\land{x\ne{y}})$$
$$x\not\le{y}\rightarrow{x\ne{y}}$$
Therefore:
$$x\not\le{y}\not\rightarrow{{(x\ge{y})}}$$
