The conditional probability is defined as:
$$P(A|B) = \frac {P(A \cap B)} {P(B)}$$
given that $$P(B) \neq 0$$
This is achieved based on our intuition, along with the Venn diagram description of the sample space $\Omega$, the event $A$, the event $B$ and the intersection of $A$ and $B$.
My question is: why can't we treat the conditional probability as the corollary of the axioms of the probability theory, instead of treating it as a separate definition? Additionally, based on this point of view, it is seen that the Bayesian rule is almost as trivial as a definition.
My takeaway is, for any event $X$, defined on the probability space $(\Omega, \sigma, P)$, $P(X)$ could always be defined as a conditional probability:
$$P(X) \equiv P(X | \Omega)$$
Am I on the right track, or am I missing something important? Thanks!