2 ordered field theorems that need proving

My textbook has kindly decided to present twotheorems without proof, meanwhile all others have proofs. I don't want to come up with a flawed proof, so I need some help.

The first of the two involved proving that 0 =/= 1

I did this one, with the help of mhf.

The remaining theorem will probably incorporate the latter into its proof...

Def. a < b <=> a<=b and a =/=b

Thm. For any ordered field, 0 < 1.

Proof:

I'm not really sure where to start, contradiction seems like it's the way to go...

Suppose that for some ordered field, 1 <= 0

0 =/= 1, so 1 < 0

by another theorem that i've proved already, (-0) < (-1)

but (-0) = 0 < (-1)

and... i feel like i'm going in circles, please help

Q.E.D.