# Metamath Proof Explorer

## Theorem mtpor

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor , one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if ph is not true, and ph or ps (or both) are true, then ps must be true". An alternate phrasing is: "once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth". -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016) (Proof shortened by Wolf Lammen, 11-Nov-2017)

Ref Expression
Hypotheses mtpor.min ${⊢}¬{\phi }$
mtpor.max ${⊢}\left({\phi }\vee {\psi }\right)$
Assertion mtpor ${⊢}{\psi }$

### Proof

Step Hyp Ref Expression
1 mtpor.min ${⊢}¬{\phi }$
2 mtpor.max ${⊢}\left({\phi }\vee {\psi }\right)$
3 2 ori ${⊢}¬{\phi }\to {\psi }$
4 1 3 ax-mp ${⊢}{\psi }$