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Theorem empty-surprise2 48796
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 48795. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1565); 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 48802. (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 48795 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wfal 1549  wex 1777  wcel 2103  wral 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2173
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-ral 3064
This theorem is referenced by: (None)
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