Number of Non-Dividing Primes Less than n is Less than Euler Phi Function of n

Theorem
Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\map w n$ denote the number of primes strictly less than $n$.

Let $\map \phi n$ denote the Euler $\phi$ function of $n$.

Then:
 * $\map w n < \map \phi n$

Proof
Let $P = \set {p < n: p \text { prime} }$.

Let $Q = \set {0 < q < n: q \perp n}$, where $q \perp n$ denotes that $q$ and $n$ are coprime.

Let $p \in P$.

From Prime iff Coprime to all Smaller Positive Integers, $p$ is coprime to $n$.

That is:
 * $p \in Q$

So, by definition of subset:
 * $P \subseteq Q$

From Integer is Coprime to 1:
 * $1 \in Q$

But as One is not Prime:
 * $1 \notin P$

Thus $P \subsetneq Q$ and so:
 * $\card P < \card Q$

By definition of $\map w n$:
 * $\card P = \map w n$

and by definition of Euler $\phi$ function:
 * $\card Q = \map \phi n$

Hence the result.