Proof that stochastic discount factor is positive in complete markets

I am looking for a reference (with a proof possibly) to understand why the completeness of markets implies that the stochastic discount factor is strictly positive in the context of intertemporal consumption, i.e. when a agent FOC implies

$$p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right]$$

where $$p_t$$ is the price of an asset giving $$x_{t+1}$$ in the following period.

For simplicity, assume agent's consumption is $$\omega$$-contingent, where $$\omega$$ belongs to a finite set $$\Omega$$. Then the FOC $$p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right]$$ becomes $$p_t = \sum_{\omega \in \Omega} \beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot x_{t+1} (\omega) \cdot P(\omega).$$ The quantity $$\beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot P(\omega)$$ can be viewed as the price of a claim with payoff $$1_{\{ \omega' =\omega\}}(\omega')$$ (i.e. a digital option that pays 1 if $$\omega$$ realizes and 0 otherwise). Such claims are exactly the Arrow-Debreu securities in the market of $$\omega$$-contingent claims.
(If the payoff $$x_{t+1}$$ is contingent on, say, state $$(\omega, \nu)$$, where agent's consumption is invariant with respect to $$\nu$$, then market is complete only with respect to $$\omega$$-contingent claims. Claims whose payoffs depend on $$\nu$$ cannot be replicated, i.e. market is not complete with respect to such claims. In the FOC, the $$\nu$$-coordinate would be integrated out.)
In equilibrium, the FOC becomes the asset pricing equation $$p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} p_{t+1}\right].$$ This means, in the market of AD securities of $$\omega$$-contingent claims, the (representative) agent holds the portfolio that is the asset itself.