I'm having some issues with the definition of the $<$ sign in set theory.
I know that more or less $\leq$ and $<$ correspond to orders and when working with $\omega$ we've shown that $n\leq m \Leftrightarrow n \in m \text{ or } n=m$. Then, we are defining the cardinal number as $X\sim n \Leftrightarrow $ there is a bijection $f:X \rightarrow n$.
Then at some point, we prove that for $m\neq n$ we have that $m$ and $n$ have different cardinalities by considering $m<n$. How can we define the strict order on $\omega$?