Negation of specific statement Definition: We say that $X\subset \mathbb{R}$ is bounded above if $\exists C\in \mathbb{R}$ s.t. $\forall x\in X$ we have $x\leq C$.
$(X \ \text{is bounded above}):=\exists C\in \mathbb{R}((\forall x\in X)\Rightarrow (x\leq C))$
So I want to take the negation of it: $$\neg(X \ \text{is bounded above})\equiv \neg(\exists C\in \mathbb{R}((\forall x\in X)\Rightarrow (x\leq C)))\equiv$$ $$\equiv\forall C\in \mathbb{R} \neg((\forall x\in X)\Rightarrow (x\leq C))$$
But we know that $\neg (P\Rightarrow Q)\equiv P\land \neg Q$ which implies that $$\neg(X \ \text{is bounded above})\equiv \forall C\in \mathbb{R} ((\forall x\in X)\land (x> C)) $$
But I know that the correct negation should be the following: $X\subset \mathbb{R}$ is not bounded above if $\forall C\in \mathbb{R}$ $\exists x\in X$ s.t. $x>C$.
What am I doing wrong in the above?
 A: Right in the begining on the second line. The big quantifier $\forall$ is not part of the implication. It is:
$\exists C\in R (\forall x\in X (x\leq C))$
Edit: Expression $(\forall x\in X) \rightarrow (x\leq C) $ isn't meaningful expression because $(\forall x\in X)$ has no logical value (it is not true-false statement) it is just quantification.
A: If we write definition as
$$(\exists C\in \mathbb{R})(\forall x\in X)(x\leq C)\quad (1)$$
Then negation gives
$$(\forall C\in \mathbb{R})(\exists x\in X)(x> C)\quad (2)$$
Now step by step: using, that $(\forall x\in X)Q(x)$ is same as $(\forall x)(x\in X \Rightarrow Q(x))$ and $(\exists x\in X)Q(x)$ is same as $(\exists x)(x\in X \land Q(x))$, definition $(1)$ can be decoded as
$$(\exists C)\big(C\in \mathbb{R} \land (\forall x)(x\in X \Rightarrow x\leq C)\big)\quad (1)$$
Taking negation gives
$$(\forall C)\big(C\notin \mathbb{R} \lor (\exists x)(x\in X \land x> C)\big)$$
which is
$$(\forall C)\big(C\notin \mathbb{R} \lor (\exists x\in X)( x> C)\big)$$
$$(\forall C)\big(C\in \mathbb{R} \Rightarrow (\exists x\in X)( x> C)\big)$$
$$(\forall C\in \mathbb{R})(\exists x\in X)(x> C)\quad (2)$$
