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Theorem empty-surprise2 47316
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 47315. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1570); 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 47322. (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 47315 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wfal 1554  wex 1782  wcel 2107  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-ral 3062
This theorem is referenced by: (None)
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