De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Forward Implication

Theorem

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

Proof

 * align="right" | 2 ||
 * align="right" | 1
 * $\neg p$
 * $\land \mathcal E_1$
 * 1
 * align="right" | 3 ||
 * align="right" | 1
 * $\neg q$
 * $\land \mathcal E_2$
 * 1
 * $\neg q$
 * $\land \mathcal E_2$
 * 1


 * align="right" | 6 ||
 * align="right" | 1, 5
 * $\bot$
 * $\neg \mathcal E$
 * 5, 2
 * 5, 2


 * align="right" | 8 ||
 * align="right" | 1, 7
 * $\bot$
 * $\neg \mathcal E$
 * 7, 3
 * align="right" | 9 ||
 * align="right" | 1, 4
 * $\bot$
 * $\lor \mathcal E$
 * 4, 5-6, 7-8
 * align="right" | 10 ||
 * align="right" | 1
 * $\neg \left({p \lor q}\right)$
 * Proof by Contradiction
 * 4, 9
 * }
 * align="right" | 1
 * $\neg \left({p \lor q}\right)$
 * Proof by Contradiction
 * 4, 9
 * }
 * }