Theorem empty-surprise2 50620

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 50619. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1598); 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-rals 50626. (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 50619	1 ⊢ ∀𝑥 ∈ 𝐴 ⊥

This proof depends on syntax axioms:  ¬ wn 3  ⊥wfal 1582  ∃wex 1812   ∈ wcel 2146  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-ral 3082

This theorem is used by: (None)