How can an asymmetric relation be antisymmetric?

Question

Let R be asymmetric.

So we have:

1. Assumption: (x,y)∈R⟹(y,x)∉R

We need to show R is antisymmetric, i.e.

2. (x,y)∈R∧(y,x)∈R⟹x=y

Since 2 is a conditional, we can assume

3. Assumption:(x,y)∈R∧(y,x)∈R

And try to show

4. x=y.

But if we simplify LHS of 3 and use Modus Ponens on (x,y)∈R and 1, our assumption, we have

5. (y,x)∉R

Which contradicts RHS of 3.

So how could an asymmetric relation be antisymmetric? Did I do something wrong here?

Answer 1

It's vacuously true! The point is that $(x,y) \in R$ and $(y,x) \in R$ can never happen, so whatever you say about it is true.

For example the statement "If $(x,y) \in R$ and $(y,x) \in R$, then $5=7$" is true.