Disjunction with Contradiction/Proof 1

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Theorem

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


Proof

By the tableau method of natural deduction:

$p \lor \bot \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor \bot$ Premise (None)
2 2 $p$ Assumption (None)
3 3 $\bot$ Assumption (None)
4 3 $p$ Rule of Explosion: $\bot \EE$ 3
5 1 $p$ Proof by Cases: $\text{PBC}$ 1, 2 – 2, 3 – 4 Assumptions 2 and 3 have been discharged

$\Box$


By the tableau method of natural deduction:

$p \vdash p \lor \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $p$ Premise (None)
2 1 $p \lor \bot$ Rule of Addition: $\lor \II_1$ 1

$\blacksquare$