ZFC and axiom of power set It seems that I do not understand thoroughly the axioms of ZFC. 
I am thinking of "is really the axiom of power set independent from the other axioms, and if it is, how to prove that?".
In other words, how to prove that ZFC without the axiom of power set is not equal to ZFC?

Consider the set $H(\kappa)$ consisting of sets $x$ which satisfy $|\text{tc}(x)| < \kappa$. As mentioned in this answer, this set $H(\kappa)$ is a model of all axioms of ZFC except power set. In fact, if one takes $\kappa$ to be a successor cardinal (such as $\aleph_1$), then one can verify that the axiom of power set if false in $H(\aleph_1)$.
Now it follows that the axiom of power set cannot be proven from the other axioms of ZFC, for if this were the case, the power set axiom would be true in $H(\aleph_1)$.
