What are the assumptions of the logical/Mathematical proof-making system?

Question

I know that axioms define a set of legal actions that we can perform on statements, by which those statements transform into other statements that are necessarily true.

Suppose that $s_1$ is an statement that its truthfulness is to be determined (aka get proved true). Then I do this:

1. I apply axiom $a_1$ on $s_1$ to get statement $s_2$. $s_2$ is unknown to be true.
2. Apply $a_2$ to get $s_3$, which its truthfulness is still unknown.
3. ...
4. Apply $a_n$ to get $s_0$, but now $s_0$ is known to be true.

Is this how proofs work? Can I now call it a Q.E.D?

In other words, is the underlying assumption of the logical/mathematical proving system this:

• A statement $s_i$ is true if it can reach any statement that we know is true?

In other words (2):

• if $\mathcal{T}$ is the exhaustive set of true statements that we know that they are true,
• and if $\mathcal{S}_i = \{s_{i+1}, s_{i+2}, \ldots, s_{i+n}\}$ is the set of possible statements that we could get from $s_i$ after applying an indefinite number of axioms on,

then can we say that $s_i$ is proven only if $\mathcal{S}_i \cap \mathcal{T} \ne \emptyset$? (else, we don't know if $s_i$ is true).

In other words (3):

• A false statement, $f_i$ can never reach a true statement, no matter how many times we apply our axioms on. So $\mathcal{F}_i \cap \mathcal{T} = \emptyset$.

Is what I am saying correct? Is that the fundamental thinking used in proofs?