# Negation of the following uniqueness quantifier expression

I had to negate the following expression:
$\underset{A\subseteq\mathbb{N}}\forall \, \underset{m\in\mathbb{N}}{\exists!} [m\in A \land \underset{a\in A}\forall m\leq a]$

Surely I expanded the quantifier and worked it out step for step and got:
$\underset{A\subseteq\mathbb{N}}\exists \, \underset{m\in\mathbb{N}}\forall [(m\in A \land \underset{a\in A}\forall m\leq a) \Rightarrow \underset{n\in\mathbb{N}}\exists((n\in A \land \underset{a\in A}\forall n\leq a) \land m\neq n)]$

So my question is if this is correct? I used 1.21 (p. 18) of this lecutre http://www.math.lmu.de/~philip/publications/lectureNotes/calc1_forInfAndStatStudents_new.pdf

The correction of my solution said that the antecedent was wrong, but I don't really get it why it should be wrong as per definition in the lecture notes.