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Theorem empty-surprise2 44812
Description: "Prove" that false is true when using a restricted "for all" over the empty set, to demonstrate that the expression is always true if the value ranges over the empty set.

Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. We proved the general case in empty-surprise 44811. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1556); the problem is that restricted "for all" works in ways that might seem counterintuitive to the inexperienced when given an empty set. Solutions to this can include requiring that the set not be empty or by using the allsome quantifier df-alsc 44818. (Contributed by David A. Wheeler, 20-Oct-2018.)

Hypothesis
Ref Expression
empty-surprise2.1 ¬ ∃𝑥 𝑥𝐴
Assertion
Ref Expression
empty-surprise2 𝑥𝐴

Proof of Theorem empty-surprise2
StepHypRef Expression
1 empty-surprise2.1 . 2 ¬ ∃𝑥 𝑥𝐴
21empty-surprise 44811 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wfal 1540  wex 1771  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-ral 3140
This theorem is referenced by: (None)
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