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Theorem mtpor 1316
 Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1317, 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 is not true, and or (or both) are true, then must be true." An alternative 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.)
Hypotheses
Ref Expression
mtpor.min
mtpor.max
Assertion
Ref Expression
mtpor

Proof of Theorem mtpor
StepHypRef Expression
1 mtpor.min . 2
2 mtpor.max . . 3
32ori 642 . 2
41, 3ax-mp 7 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 629 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  mtpxor  1317  ordtriexmid  4247
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