In class we had the Proposition about density of compactly supported continuous functions $C_c(X)$ in $L^p(X)$

(If you do not know the Prop. see e.g.: https://planetmath.org/compactlysupportedcontinuousfunctionsaredenseinlp)

My confusion is about the sometimes used notation of density:

$A$ is dense in $X:\iff\bar A = X$

If we use this notation together with twice the Propoistion we get:

$L^p(X) = \overline {C_c(X)} = L^q(X)$ for $1\leq p,q < \infty$

and therefore

$L^p(X) = L^q(X)$ for $1\leq p,q < \infty$

But equality is not given in general (e.g. $L^p$ and $L^q$ space inclusion)

So I guess that the equal symbol in this dense notaion is not the equal symbol itself?