Theorem mtpor 1695

Description: Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1696, 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 𝜑 is not true, and 𝜑 or 𝜓 (or both) are true, then 𝜓 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	⊢ ¬ 𝜑
mtpor.max	⊢ (𝜑 ∨ 𝜓)

Assertion

Ref	Expression
mtpor	⊢ 𝜓

Proof of Theorem mtpor

Step	Hyp	Ref	Expression
1		mtpor.min	. 2 ⊢ ¬ 𝜑
2		mtpor.max	. . 3 ⊢ (𝜑 ∨ 𝜓)
3	2	ori 390	. 2 ⊢ (¬ 𝜑 → 𝜓)
4	1, 3	ax-mp 5	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   ∨ wo 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385

This theorem is referenced by:  mtpxor  1696  tfrlem14  7487  cardom  8812  unialeph  8924  brdom3  9350  sinhalfpilem  24215  mof  32409  dvnprodlem3  40163