proof that 0 is not equal to 1

I'm not very experienced when it comes to writing proofs, and I have come up with two proofs for 0 =/= 1, both by reductio ad absurdum. Is either one of them valid?

for any field F, 0 denotes the additive identity and 1 denotes the multiplicative identity

the real numbers are a set

sets are a collection of distinct elements

if 0 = 1 then 0 is not in the real numbers xor 1 is not in the real numbers

if 0 xor 1 are not in the real numbers then, the real numbers are no longer a field

but the real numbers are a field

therefore 0 =/= 1

a is a real number

suppose 0 = 1, then a + 0 = a and a + 1 = a

the real numbers are a field

there exists an element 0 for any a in a field F such that a + 0 = a

therefore 0 =/= 1