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

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

By the tableau method of natural deduction:

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

$\blacksquare$


Sources