# Disjunction with Contradiction/Proof 1

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## Theorem

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

## Proof

By the tableau method of natural deduction:

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:

Line | Pool | Formula | Rule | Depends upon | Notes | |
---|---|---|---|---|---|---|

1 | 1 | $p$ | Premise | (None) | ||

2 | 1 | $p \lor \bot$ | Rule of Addition: $\lor \II_1$ | 1 |

$\blacksquare$