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

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Theorem

$p \lor q \vdash \neg \paren {\neg p \land \neg q}$


Proof

By the tableau method of natural deduction:

$p \lor q \vdash \neg \paren {\neg p \land \neg q} $
Line Pool Formula Rule Depends upon Notes
1 1 $p \lor q$ Premise (None)
2 2 $\neg p \land \neg q$ Assumption (None)
3 2 $\neg p$ Rule of Simplification: $\land \EE_1$ 2
4 2 $\neg q$ Rule of Simplification: $\land \EE_2$ 2
5 5 $p$ Assumption (None)
6 2, 5 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 5, 3
7 7 $q$ Assumption (None)
8 2, 7 $\bot$ Principle of Non-Contradiction: $\neg \EE$ 7, 4
9 1, 2 $\bot$ Proof by Cases: $\text{PBC}$ 1, 5 – 6, 7 – 8 Assumptions 5 and 7 have been discharged
10 1 $\neg \paren {\neg p \land \neg q}$ Proof by Contradiction: $\neg \II$ 2 – 9 Assumption 2 has been discharged

$\blacksquare$


Sources