If a property P is true for some arbitrary element in a given set, then why is true for all the elements in the set? Here is an example.

Theorem. Suppose $A\subset B$ and $B\subset C$. Then $A \subset C$.

Proof. Let $x \in A$. Since $A\subset B$, it follows that $x \in B$. And yet again since $B \subset C$, we can conclude $x \in C$, hence $A \subset C$. $\blacksquare$

Can anyone explain it to me why it holds true?
First, a quick terminological note: the specific rule of formal logic which is allowing you to do this is universal generalization. However, just knowing there's a rule for it probably won't help your intuition ...



It may be helpful to think of a proof in terms of a game: you claim some fact, and your opponent is trying to show that it's false. A proof is really a winning strategy for you in this game - that is, a proof is a way to withstand any attempt to show that you're wrong.

In this case, you claim


  Every element of $A$ is in $C$.


What's your opponent to do? Well, they need to find some $x$ which is in $A$ but not in $C$. So they bring you some $x$, and tell you "$x$ is in $A$ and-"

"Stop," you say - "I now know (or rather, can show) that $x$ is in $C$. So your attempt failed."

In this case, I think the crucial intuitive feature is: your opponent brings you $x$. You don't get to choose $x$; rather, you have to be able to handle whatever challenge your opponent makes. Proving a result for an arbitrary $x$ is exactly that - showing that no matter what $x$ you are presented with, you can still "win." Thinking of it this way, in my experience, helps get rid of the impression that you're somehow "cheating" (and keeping track of the roles of you/your opponent can be helpful in analyzing more complicated proof structures).

(This kind of interpretation of proofs - and of mathematical truth - is something I've found very useful in clarifying many concepts; the key phrase here is "game semantics.")



A final note: getting rid of the language of games, what's really going on here is that you've shown that there is no counterexample to your claim. This is really what a universal claim like "$A\subset C$" (in words, "for all $x$, if $x\in A$ then $x\in C$" - that "for all" is why it's a universal claim) means.
We can go this way, the probability for the given property to be true for an arbitrary element in the set is 1 coz it's true for that arbitrary element. Now since it's an "arbitrary element" the probability of anything being true for any element in the set is equal to that of the initial element taken till there is  uniformity within the set. Hence the property will be true for all elements