The sources is this notes http://www.maths.manchester.ac.uk/~jeff/lecture-notes/MATH33001.pdf
The thing is I'm confused about the proof of MON.
We know that if $\Gamma | \theta$, then $\Gamma \cup \Delta | \theta$
So to prove it you need to say let $\Gamma \vdash \theta$
So exists a proof $\Gamma_1 | \theta_1, \Gamma_2 | \theta_2 ,...,\Gamma_k | \theta_k$ where $\Gamma_i \subset \Gamma$ and $\theta_k=\theta$
$\Gamma_1 | \theta_1, \Gamma_2 | \theta_2 ,...,\Gamma_k | \theta_k, \Gamma_k \cup \Delta_1 | \theta$
But, does $\Delta_1$ have to be finite. Also, what the point of this anyway? Having a discussion that says the point is extending it to infinity. However, I think just think you can give a proof in the form of finite subset.