MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  celaront Unicode version

Theorem celaront 2258
Description: "Celaront", one of the syllogisms of Aristotelian logic. No  ph is  ps, all  ch is  ph, and some  ch exist, therefore some  ch is not  ps. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
celaront.maj  |-  A. x
( ph  ->  -.  ps )
celaront.min  |-  A. x
( ch  ->  ph )
celaront.e  |-  E. x ch
Assertion
Ref Expression
celaront  |-  E. x
( ch  /\  -.  ps )

Proof of Theorem celaront
StepHypRef Expression
1 celaront.maj . 2  |-  A. x
( ph  ->  -.  ps )
2 celaront.min . 2  |-  A. x
( ch  ->  ph )
3 celaront.e . 2  |-  E. x ch
41, 2, 3barbari 2257 1  |-  E. x
( ch  /\  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator