Consider a cdf $F:[0,1]\to [0,1]$ such that $F(x)\leq x$ for all $x\in [0,1]$. Is this a well-known property?

As the title says, I am considering a CDF $F:[0,1]\to [0,1]$ such that $F(x)\leq x$ for all $x\in [0,1]$.

This seems like a pretty basic property and I was wondering if:

1. There is a name for the CDFs that meet this property, or some reference(s) that includes characterizations
2. CDFs like $F$ meet other known properties.

One could start by noting that $x-F(x)$ is an increasing function since $F'(x)\leq 1$. Then taking x>0 and studying the hazard ratio:

$$\frac{F'(x)/x}{1/x-F(x)/x}$$

I have the intuition that it should be decreasing since $F'(x)/x\leq 1/x$ and $F(x)/x\leq 1$. But I don't want to prove it until I am sure I am going in the right direction.