Proof of Fermat's Last Theorem for n = 4

I was thinking about my last post,
where I said that it might be possible
that Fermat's Last Theorem could be
solved by elementary maths. Well, I
was partially right- I attempted a proof
for Fermat's Last Theorem's second case,
which says that the equation

X⁴ + Y⁴ = Z⁴

My Proof

has no integer values. So this is how it goes-

If there are any mistakes in the math, leave the
error in the comments.