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Theorem empty-surprise2 42338
 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 42337. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1510); 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 42344. (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 42337 1 𝑥𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ⊥wfal 1480  ∃wex 1695   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-ral 2901 This theorem is referenced by: (None)
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