*Language, Proof, and Logic* Fitch Proof Exercise 6.16

Question

This is the last proof I need to finish. I've really been struggling with this one even though it seems so simple. Instructions say use Tarski's world if the sentences are consistent (they aren't), or use Fitch to prove they're inconsistent. Obviously a and b can't be the same size when previously stating that one of the two has to be smaller. Any help would be much appreciated. Says ana con is allowed. Problem below..

Premises:

$\operatorname{Smaller}(a, b) \lor \operatorname{Smaller}(b, a),$

$\operatorname{SameSize}(a, b)$

Goal conclusion: $⊥$

Here's a photo of the blank one , proof

Here's what I've done (wrong), proof2

Answer 1

$\operatorname{SameSize}(a, b)$ implies $\operatorname{SameSize}(b,a)$

$\operatorname{Smaller}(a, b)$ implies $\lnot \operatorname{SameSize}(a, b)$ and $\operatorname{Smaller}(b, a)$ implies $\lnot \operatorname{SameSize}(b, a)$.

Here for the proof [see LP&L, Ex.6.16, page 164 and page 159] :

1) $\operatorname{Smaller}(a, b) \lor \operatorname{Smaller}(b,a)$

2) $\operatorname{SameSize}(a, b)$

3) $\operatorname{SameSize}(b, a)$ --- from 2) by Ana Con

--- 4) $\operatorname{Smaller}(a, b)$ --- assumed from 1)

--- 5) $\lnot \operatorname{SameSize}(a, b)$ --- from 4) by Ana Con

--- 6) $\bot$ --- $\bot$-Intro from 2) and 4)

--- 7) $\operatorname{Smaller}(b,a)$ --- assumed from 1)

--- 8) $\lnot \operatorname{SameSize}(b,a)$ --- from 7) by Ana Con

--- 9) $\bot$ --- $\bot$-Intro from 3) and 8)

10) $\bot$ --- $\lor$-Elim 1), 4)-6) and 7)-9).