I was watching [Harvey Mudd's real analysis video](https://youtu.be/AQHVdiXRXQA?t=1h6m4s) talking about the finite intersection property. The first step of the proof relies using contradiction and thus starts by saying that the intersection of all the sets is empty:

$$ \cap\_{\alpha } K\_{\alpha} = \emptyset $$

Then the professor says the following (perhaps paraphrased a little bit):

> 
> If a point is in the intersection then it means that point is not in any of the complements. So if there is no point in any intersection then some point must be in some complement.
> 
> 
> 

The first part of the statement seems obvious to me, if one is in all the sets then thats the same as saying that point must be in every set simultaneously. If one is in all the sets simultaneously then you can't be in any one of the complements (since otherwise you'd not be in that set and thus can't be in all of them simultaneously). Basically this relies that being in a set means you can't be in the complement of that set and being in all the sets means you can't be in any one of the complements. Basically:

$$ x \in \cap\_{\alpha} K\_{\alpha} \implies x \not \in\cup\_{\alpha} K^c\_{\alpha} $$

or

$$ \cap\_{\alpha} K\_{\alpha} = ( \cup\_{\alpha} K^c\_{\alpha} )^c $$

However, the argument proceeds to then say:

> 
> So if there is no point in any intersection then some point must be in some complement.
> 
> 
> 

which is the part I am struggling to understand. I wish to understand it sort of in plain english. What they call sometimes "intuitively" because that second part of the conclusion is not something that would have occurred to me and it seems something that should have flowed naturally from a train of thoughts rather than some symbol crunching game.

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The reason I say not from a "symbol crunching game" is because its quite trivial to just start applying negations and complements until you force the answer out. e.g.

$ \cap\_{\alpha } K\_{\alpha} = \emptyset $

using DeMorgan's $ (\cap\_{\alpha } K\_{\alpha})^c = \cup K^c\_{\alpha}$ and complementing it we get:

$ \cap\_{\alpha } K\_{\alpha} = (\cup K^c\_{\alpha})^c = \emptyset$

then taking the complement of everything we get:

$\cup K^c\_{\alpha} = \emptyset^c = X$

i.e. the complement of nothing is everything and then we get the desired answer. The reason I don't like this reasoning is because it just feels like symbol crunching rather the way professor Su explained it seems there is a naturally flowing way to reach that conclusion from logic or just reasoning without requiring long winding manipulation of rules and symbols. 

So what is the simple logical way to understand that part of the reasoning?

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I think I've identified what confuses me the most. It seems that DeMorgan's law's talks about the equality of two sets in terms of what they have, but the fact that in the question premise we are dealing with the **empty set** itself (i.e. a set that has nothing) is what is confusing me and then making a jump to a set that has everything coupled with DeMorgan's, complements seems to confusing to deal with all at once. I think because the set has nothing and the proofs I have for those are in terms of things that I do have, is what is confusing me. Perhaps this can bias answers to address what seems to be the crux of my confusion.