How can you disprove the statement $4=5$? I know this sounds insane. But bear with me, my friend said this to me with a straight face: 

Can you disprove the statement: $4=5$?

And I was like is that even a question, thats obvious, $5$ is $4+1$ and $5$ comes after $4$.
He was like: but you haven't still disproved my statement.
Something like this went for around 30 minutes...and he wasn't happy at the end at all
Is he correct? If not, how does one actually give a good sensible answer to such a person?
 A: Subtract 4 from both sides of $4=5$, yielding $0=1$. Now $0$ is defined to be the number such that $0+x=x$ for every $x$ and $1$ the number such that $1\cdot x = x$ for every $x$. But with that definition of $0$ we get that $0\cdot x = (0+0)\cdot x = 0\cdot x + 0\cdot x$, so $0\cdot x =0$ for every $x$. So since $1$ is the multiplicative identity (the property described above), but $0$ behaves differently multiplicatively, they can't be equal. 
A: A good sensible answer would have to be based on some assumptions, i.e. on a set of axioms. The simpler the axioms, the better, but you cannot get away from some kind of assumptions or axioms. 
In truth, this situation holds for any theorem of mathematics, not just the theorem $4 \ne 5$. In calculus, for example, the assumptions one often uses are the axioms for the real numbers; from those, one can produce a proof of any theorems you run across in a calculus text.
In simple integer arithmetic, the assumptions one often uses are the Peano Axioms. It would be straightforward, albeit perhaps a little tedious, to produce a proof that $4 \ne 5$ using the Peano axioms.
One thing that happens with proofs is that in addition to having a long tedious formal proof, you also try to boil the proof down into basic intuitive statements. I'll venture to guess that in the end, the formal proof using Peano's Axioms that $4 \ne 5$ will probably boil down to exactly what you said, "$5$ comes after $4$". 
A: Hold up 4 fingers on your left hand. Hold up 5 fingers on your right hand (yes, count your thumb as a finger). Show that no matter how you try to pair them off, there's always one left over on your right hand. For added effect, make it the middle finger. 
A: From the point of view of a non-mathematician, but one who is forced to use some degree of logical and internally consistent thoughts, arguments, and communications with other humans on a daily basis, before proving any kind of relation between any two "objects", it is usually required that an agreement exists on what these objects are, or on what we agree ("rightly" or "wrongly") that they are. So I suggest that you and your friend sit down and agree on a common definition of what the visual symbol "$4$" means/represents, what the visual symbol "$5$" means/represents, and what the visual symbol "$=$" means/represents.  
After that, I feel that you will be able to debate more fruitfully whether the relation $4=5$ is valid or not, given your agreed upon definitions.  
As an example, just last night I agreed with a friend that "$4$" represents "moving away from the center of earth's gravity", "$5$" represents "hot chocolate" and "$=$" means "not the same thing". We also have reached agreement on the meaning of all words used in the above definition. Then we were able to reach the conclusion that $4=5$ is a useful compact expression of a relation that actually holds.
A: 
Something like this went for around 30 minutes...and he wasn't happy at the end at all.
Is he correct? If not, how does one actually give a good sensible answer to such a person?

At some point in a child's life, they learn the word "why". They learn that they can basically question everything. When justification is given for an answer, that justification can be questioned as well. In these situations, a "sensible answer" doesn't exist because the person you are talking to isn't sensible.
That said, you friend might actually be sensible. So here is what I would do.
First you need to agree on a set of facts that don't need justification. A proof is something that goes from something that is know to be true to the statement you want to show is true. If you can't, for example, agree on the definition of natural numbers, then it isn't really possible to write down any proofs involving the natural numbers. So, you need to find common ground. If your friend refuses to accept anything as true, then no proof exists.
A: It's all about definitions. I suggest you review (a) the definitions of mathematical equality. "$x=y$" has different meanings depending on context -- in this case, you are dealing with (b) an equivalence relation: 

In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition. The intersection of any two different cells is empty; the union of all the cells equals the original set. These cells are formally called equivalence classes.

Since 4 and 5 are different, they occupy separate uniquely-partitioned cells in the set of integers (or of reals or rationals, etc). How do we know that 4 and 5 are different? They have different properties: 5 is prime, 4 is not; 5 is odd, 4 is not; 4 is a perfect square, 5 is not, etc. Since they are different, 4 and 5 cannot occupy the same cell in the set, so the statement $4=5$ does not satisfy the definition of an equivalence relation. 
