The two completions of $\{0,1,2\}, \{0,3,4\}, \{0,5,6\}$

Setting. The points are the seven elements of $\{0,1,2,3,4,5,6\}$, written $[7]$ below. The theorem is about an arbitrary structure $S$ of the following kind, called a Steiner triple system on $[7]$. It is given by a finite family $\mathcal{L}(S)$ of subsets of $[7]$, called lines, and two conditions:

• every line $\ell \in \mathcal{L}(S)$ has exactly $3$ elements;
• for any two distinct points $x \neq y$ in $[7]$, there is exactly one line $\ell \in \mathcal{L}(S)$ with $x \in \ell$ and $y \in \ell$.

The structure holds nothing else. The two conditions are proofs, so $S$ is determined by its family of lines $\mathcal{L}(S)$. The imported files also define a specific system called "fano" (lines $\{i, i+1, i+3\}$ mod $7$), a predicate "IsFano" and a predicate "RoleColouring". None of these appear in this statement: it uses only the notion of a Steiner triple system on $[7]$ given above.

Hypotheses. $S$ is any Steiner triple system on $[7]$, as above, such that the following three sets are all lines of $S$:

There are no other hypotheses. The hypotheses can be satisfied together: for example, the first family listed below meets all of them.

Conclusion. The whole family of lines of $S$ is exactly equal to one of the two seven-element families below. This is equality of sets of $3$-subsets: $S$ has these seven lines and no others.

or

The "or" is an ordinary (inclusive) disjunction. The two families are different sets (for example, $\{1,3,6\}$ is in the first but not in the second), so at most one of the two can hold for a given $S$. The statement says only that $\mathcal{L}(S)$ is literally one of these two labelled families. It says nothing about isomorphism, about the "fano" system, or about the number of lines beyond what these equalities imply.