simple question regarding contradicting a phrase

Question

if i have a logic property(let's call it a) that follows $\forall a,b \in \Sigma$ : $a \Rightarrow b$ or $b \Rightarrow a$ - if i want to show that $\lnot a$ still follow this property? i.e if $\Sigma$ follow a does $\lnot a$ for each $a \in \Sigma$ still follow this property?

for instance, if we contradict we get that for every $a,b \in \Sigma$ $\lnot a \Rightarrow \lnot b \lor \lnot b \Rightarrow \lnot a$ and if we contradict again we get that $a \Rightarrow b$ or $b \Rightarrow a$. is this correct as a proof?

Answer 1

In classical logic $(a \to b) \lor (b \to a)$ is a tautology, i.e. a "logical law" that is always TRUE.

Thus, obviously, also $(\lnot a \to \lnot b) \lor (\lnot b \to \lnot a)$ is always TRUE.

The simple explanation is that both are instances of the tautological schema :

$(\varphi \to \psi) \lor (\psi \to \varphi)$.