One common fallacy is the fallacy of composition, where one argues from the fact that each part of a thing has a certain feature to the conclusion that the whole thing has that feature. For instance, one could argue that every brick of the house is cube-shaped, therefore the house is cube-shaped. Or one could argue that each part of one's brain is unconscious, therefore the whole brain is unconscious. These inferences are fallacious.
However, I think it is worth noting that not all inferences from properties of parts to properties of the whole are invalid. If each part of a wall is made entirely of stone, then the whole wall is made entirely of stone. Similarly, if each part of the ball is entirely red, then the whole ball is entirely red. And so on.
Contingency seems to be like this, at least in this case. So here's an argument that the universe must be contingent:
(1) Each part of the universe is essential to the universe; in other words, *this* particular universe we live in would not be the same universe without all its parts.
(2) If each part of the universe is essential to the universe, and some part of the universe is contingent, then the whole universe is contingent.
(3) Some part of the universe is contingent.
(4) So the whole universe is contingent.
And then of course we can run the cosmological argument:
(5) Whatever is contingent requires an explanation for its existence.
(6) So the universe requires an explanation for its existence. (by 4 and 5)
It might not be immediately obvious, but premise (2) can actually be proven by the definitions of the term 'essential' and 'contingent'. So the only real premises are (1) and (3).
Let me give a proof for (2):
(A) Suppose each part of the universe is essential to the universe.
(B) And suppose some part of the universe is contingent.
(C) By (B), it is possible for some part of the universe to not exist. [Definition of 'contingency']
(D) By (A), if some part of the universe does not exist then the universe does not exist. [Definition of 'essential']
(E) By (C) and (D), it is possible for the universe not to exist; so the universe is contingent.
(F) So, If each part of the universe is essential to the universe and some part of the universe is contingent, then the universe is contingent. [by (A) through (E)]
Premise (1) might be contentious. However, let's stipulate what we mean by 'universe'; once we define this term, we can take the conclusion of the argument to hold of whatever entity we define 'universe' as denoting. Now, I understand the universe to be the sum of all space, time, energy, and matter. Under this definition, premise (1) is true, since the universe is a mereological sum.
Though I decided to define 'universe' this way, presumably it is still a substantial conclusion that this entity must have an explanation for its existence. Depending on what version of the principle of sufficient reason you hold, (5) and (6) can be modified and made stronger or weaker, giving a stronger or weaker conclusion. I think that from (4) one can actually prove the existence of God.