# Negation of Excluded Middle is False/Form 1

## Theorem

$\neg (p \lor \neg p) \vdash \bot$

## Proof

By the tableau method of natural deduction:

$\neg (p \lor \neg p) \vdash \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg (p \lor \neg p)$ Assumption (None)
2 1 $\neg p \land \neg \neg p$ Sequent Introduction 1 De Morgan's Laws
3 1 $\neg p$ Rule of Simplification: $\land \mathcal E_1$ 2
4 1 $\neg\neg p$ Rule of Simplification: $\land \mathcal E_2$ 2
5 1 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 3, 4

$\blacksquare$

## Remark

The specific form of De Morgan's Laws used in this proof does not itself rely on Law of Excluded Middle in any way.