Tautology and Contradiction

Context
Natural deduction

Theorems
A contradiction implies the negation of a tautology:

$$\bot \vdash \lnot \top$$

That is, a falsehood can not be true.

A tautology implies the negation of a contradiction:

$$\top \vdash \lnot \bot$$

That is, a truth can not be false.

A conjunction with a contradiction:

$$p \land \bot \vdash \bot$$

A disjunction with a contradiction:

$$p \lor \bot \vdash p$$

Proofs
These are proved by the Tableau method.

$$\bot \vdash \lnot \top$$:

$$\top \vdash \lnot \bot $$:

$$p \land \bot \vdash \bot$$:

$$p \lor \bot \vdash p$$: