Transitiveness of set sizes Given that: $$|A|\le|B|<|C|$$ Prove that: $$|A|<|C|$$ * * * I proved that: $$|A|\le|C|$$ By showing a $1:1$ function from $A$ to $C$, in the following way: $$\exists f:A\to B, \exists g:B\to C$$ $f$ and $g$ are $1:1$, So their composition, $g \circ f$ is $1:1$ too. Now I need to show that no onto function exists from $A$ to $C$.

Suppose $f:A\to C$ is a surjection and let $g:A\to B$ be an injection. $g$ has a partial inverse $g^{-1}:g(A)\to A$ and then $f\circ g^{-1}:g(A)\to C$ is onto. If we arbitrarily extend $f\circ g^{-1}$ to $B$, then we get a surjection from $B$ onto $C$, contradiction.