Theorem empty-surprise2 49823

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 49822. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1569); 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 49829. (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 49822	1 ⊢ ∀𝑥 ∈ 𝐴 ⊥

Syntax hints:  ¬ wn 3  ⊥wfal 1553  ∃wex 1780   ∈ wcel 2111  ∀wral 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-ral 3048

This theorem is referenced by: (None)