What is the mistake in this proof? During a long night without sleep I managed to come up with a proof for a statement I know is false, and for the life of me I cannot figure out what I did wrong. Where is my mistake?

Theorem: Let $f:A\to B$ be any function. There is a bijection from A onto B.
Proof: By contradiction. Assume there is no bijection from A onto B. Consider the statement "If f is a surjection, then f is an injection." This statement is true iff either f is not a surjection or f is an injection (this is material implication). But this statement is false, because if f is a surjection it can't be an injection because there are no bijections. So then the statement "Either f is not a surjections or it is an injection" is false, meaning f is a surjection and not an injection (demorgan). But if f is not an injection then either f is not an injection or it is a surjection. Using material implication again, this is equivalent to the statement "if f is an injection then it is a surjection," which is false because then there would be a bijection. So f is an injection and not a surjection, which is clearly a contradiction.

This is obviously absurd but I'm having trouble thinking straight. Where is my error?

Hint: try a simple example: e.g., let $A = \{1\}$ and $B = \{1, 2\}$, then there is a function from $A$ to $B$, e.g., the function $f$ that maps $1$ to $1$. Now work through your argument to see where it breaks down.
