The MTW Reformulated
1. Necessarily, two possible propositions that do not yield a contradiction when in conjunction with one another are compossible. (Premise)
2. Possibly, something presently exists. (Premise)
3. Possibly, there was a past time at which nothing contingent existed. (Premise)
4. Necessarily, if there was a past time at which nothing existed, then nothing would presently exist. (Premise)
5. The conjunction of (2) and (3) does not yield a contradiction. (Premise)
6. Possibly, something necessary exists. (Implied by 1 - 5)
7. Therefore, something necessary exists. (From 6 and S5)
The Argument's Validity
Assume (8): something necessary does not exist. (8) and S5 imply (9): it is not possible for something necessary to exist.
However, we know from (1) and (5) that (10): "something presently exists" and "there was a past time at which nothing contingent existed" are compossible. Let's call this possible world w1. In conjunction with (4), it follows from (10) that (11): there was never a past time in w1 at which nothing existed. Since the only remaining entity that could exist at the time that all contingent things failed to exist is something necessary, it follows that (12): something necessary exists in w1. Because (12) contradicts (9), we may conclude that (9), as well as (8) by extension, are false. Therefore, something necessary exists.