Theorem empty-surprise2 50142

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 50141. 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 50148. (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 50141	1 ⊢ ∀𝑥 ∈ 𝐴 ⊥

Syntax hints:  ¬ wn 3  ⊥wfal 1554  ∃wex 1781   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-ral 3053

This theorem is referenced by: (None)