Conjunction with Contradiction/Proof 1

Theorem

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

Proof

By the tableau method of natural deduction:

$p \land \bot \vdash \bot$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \land \bot$	Premise	(None)
2		1	$\bot$	Rule of Simplification: $\land \EE_2$	1

$\Box$

By the tableau method of natural deduction:

$\bot \vdash p \land \bot$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$\bot$	Premise	(None)
2		1	$p \land \bot$	Rule of Explosion: $\bot \EE$	1	From a bottom, we can prove what we like

$\blacksquare$