What is the intuitive explanation for the CDF of any random variable to follow uniform distribution (0,1)? If $X$ is a continuously distributed random variable and $F$ is the c.d.f. of its distribution, then $F(X)$ is uniformly distributed in the interval $(0,1).$

While i'm clear with the mathematical proof, I'm looking for an intuitive answer to this.
Assuming we are dealing with continuous random variables


the CDF $F(X)$ is a continuous increasing function of $X$ where the probability that $F(X) \le p$  for $p \in (0,1)$ is the probability that $X \le q$ when $F(q)=p$, which is $p$
so $F(X)$ has a uniform distribution on the unit interval, since the probability it is less than or equal to a given $p \in (0,1)$ is $p$
Assuming we are dealing with continuous random variables


the CDF $F(X)$ is a continuous increasing function of $X$ where the probability that $F(X) \le p$  for $p \in (0,1)$ is the probability that $X \le q$ when $F(q)=p$, which is $p$
so $F(X)$ has a uniform distribution on the unit interval, since the probability it is less than or equal to a given $p \in (0,1)$ is $p$