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

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Theorem

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


Proof

By the tableau method of natural deduction:

$\neg p \lor \neg q \vdash \neg \left({p \land q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \lor \neg q$ Premise (None)
2 2 $p \land q$ Assumption (None)
3 2 $p$ Rule of Simplification: $\land \mathcal E_1$ 2
4 2 $q$ Rule of Simplification: $\land \mathcal E_2$ 2
5 5 $\neg p$ Assumption (None)
6 2, 5 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 3, 5
7 7 $\neg q$ Assumption (None)
8 2, 7 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 4, 7
9 1, 2 $\bot$ Proof by Cases: $\text{PBC}$ 1, 5 – 6, 7 – 8 Assumptions 5 and 7 have been discharged
10 1 $\neg \left({p \land q}\right)$ Proof by Contradiction: $\neg \mathcal I$ 2 – 9 Assumption 2 has been discharged

$\blacksquare$


Sources