Theorem empty-surprise2 46487

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 46486. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1567); 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 46493. (Contributed by David A. Wheeler, 20-Oct-2018.)

Hypothesis

Ref	Expression
empty-surprise2.1	⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴

Assertion

Ref	Expression
empty-surprise2	⊢ ∀𝑥 ∈ 𝐴 ⊥

Proof of Theorem empty-surprise2

Step	Hyp	Ref	Expression
1		empty-surprise2.1	. 2 ⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴
2	1	empty-surprise 46486	1 ⊢ ∀𝑥 ∈ 𝐴 ⊥

Syntax hints:  ¬ wn 3  ⊥wfal 1551  ∃wex 1782   ∈ wcel 2106  ∀wral 3064

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 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-ral 3069

This theorem is referenced by: (None)