Proof of $LP$ is in $coNP$ without showing it is in $P$?

Note: all vector inequalities in this reply are to be interpreted pointwise.

Given a linear feasibility problem, you can always rewrite it in the following canonical form: given a matrix $A in mathbb{R}^{m times n}$ and vector $b in mathbb{R}^m$, does there exist an $x geq vec{0}$ such that $Ax leq b$?

Farkas' lemma states that when a system of inequalities such as the above has no solution, then there exists a "witness" vector $y in mathbb{R}^{m}$ such that

1. $y geq vec{0}$,
2. $A^{intercal}y geq vec{0}$, and
3. $b^{intercal}y < 0$.

Effectively, $b$ gives you a nonnegative scaling factor for each constraint, such that when you add up all of the scaled constraints, the left-hand side ends up with a sum of a bunch of nonnegative terms, and the right-hand side ends up with a strictly negative value. That is, $y$ provides a direct way to prove that the inequalities are inherently contradictory.

Therefore, the existence of such a $y$ is both a necessary and sufficient condition for any given problem instance to be infeasible, meaning that it can be used as a polynomial-time checkable certificate of infeasibilty. So linear feasibility checking is in co-NP.