Strict Ordering can be Expanded to Compare Additional Pair/Proof 1

Proof
First, note that since $\prec$ is a strict ordering, $\preceq$ is an ordering by Reflexive Closure of Strict Ordering is Ordering.

$a$ and $b$ are $\preceq$-incomparable
Suppose that $a \preceq b$.

By the definition of reflexive closure, either $a \prec b$ or $a = b$.

Each possibility contradicts one of the premises, so $a \not \preceq b$.

For the same reasons, $b \not \preceq a$.

$\prec'^+$ is antireflexive
Let $p \in S$.

Then by the definition of strict ordering:

$p \not\prec p$.

Suppose that $p \preceq a$ and $b \preceq p$.

Then since $\preceq$ is transitive, $b \preceq a$, contradicting the fact that $a$ and $b$ are $\preceq$-incomparable.

Since neither $p \prec p$ nor $\paren {p \preceq a} \land \paren {b \preceq p}$ holds:
 * $p \not \prec'^+ p$.

Since this is the case for all $p \in S$, $\prec'^+$ is antireflexive.

$\prec'^+$ is transitive
Let $p \prec'^+ q$ and $q \prec'^+ r$.

Then there are three possibilities:

$(1): \quad p \prec q$ and $p \prec r$

Because $\prec$ is transitive:
 * $p \prec r$

Thus:
 * $p \prec'^+ r$

$(2): \quad p \prec q$, $q \preceq a$, and $b \preceq r$

By Extended Transitivity, $p \prec a$.

Since $p \prec a$ and $b \prec r$:
 * $p \prec'^+ r$

$(3): \quad p \preceq a$, $b \preceq q$, and $q \prec r$

By Extended Transitivity:
 * $b \prec r$

Therefore:
 * $b \preceq r$

Since $p \preceq a$ and $b \preceq r$:
 * $p \prec'^+ r$

Note that it is impossible to have $p \preceq a$, $b \preceq q$, $q \preceq a$ and $b \preceq r$.

If that were so, $b \preceq q$ and $q \preceq a$ together would imply by transitivity that $b \preceq a$.

But this contradicts the fact that $a$ and $b$ are $\preceq$-incomparable.

Thus in all cases, $p \prec'^+ q$ and $q \prec'^+ r$ imply $p \prec'^+ r$, so $\prec'^+$ is transitive.

Since $\prec'^+$ is transitive and antireflexive, it is by definition a strict ordering.

$\prec'^+$ is the transitive closure of $\prec'$
First note that $\prec'$ is a subset of $\prec'^+$:

If $p \prec' q$ then either
 * $p \prec q$ or
 * $p = a$ and $q = b$

If $p \prec q$ then $p \prec'^+ q$ by definition.

If $p = a$ and $q = b$, then $p \preceq a$ and $q \preceq b$ by the definition of $\preceq$.

Thus $p \prec'^+ q$ by the definition of $\preceq'^+$.

Let $\RR$ be a transitive relation containing $\prec'$.

Let $p, q \in S$ and let $p \prec'^+ q$.

Then either:
 * $p \prec q$ or
 * $p \preceq a$ and $b \preceq q$

If $p \prec q$, then by the definition of subset, $p \mathrel \RR q$.

Suppose instead that $p \preceq a$ and $b \preceq q$. By the definition of $\preceq'$, $a \preceq' b$.

Since Reflexive Closure is Closure Operator, it is increasing, so $\preceq \subseteq \RR^=$.

Thus $p \mathrel \RR^= a$, $a \mathrel \RR^= b$, and $b \mathrel \RR^= q$.

Since $\RR$ is transitive, $\RR^=$ is as well, by Reflexive Closure of Transitive Relation is Transitive.

Thus $p \mathrel \RR^= q$.

Since $\preceq'^+$ is antireflexive and we assumed $p \preceq'^+ q$, we conclude that $p \ne q$.

Thus by the definition of reflexive closure:
 * $p \mathrel \RR q$

Thus we have shown that $\preceq'^+$ is a subset of $\RR$.

Since this holds when $\RR$ is any transitive superset of $\preceq'$, $\preceq'^+$ is the transitive closure of $\preceq'$.