# Absorption Laws (Logic)/Conjunction Absorbs Disjunction/Proof 2

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## Theorem

$p \land \left({p \lor q}\right) \dashv \vdash p$

## Proof

By calculation:

 $\displaystyle p \land \left({p \lor q}\right)$ $=$ $\displaystyle \left({p \lor \bot}\right) \land \left({p \lor q}\right)$ Disjunction with Contradiction $\displaystyle$ $=$ $\displaystyle p \lor \left({\bot \land q}\right)$ Disjunction is Left Distributive over Conjunction $\displaystyle$ $=$ $\displaystyle p \lor \bot$ Conjunction with Contradiction $\displaystyle$ $=$ $\displaystyle p$ Disjunction with Contradiction

$\blacksquare$