Theorem empty-surprise2 49762

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 49761. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1568); 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 49768. (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 49761	1 ⊢ ∀𝑥 ∈ 𝐴 ⊥

Syntax hints:  ¬ wn 3  ⊥wfal 1552  ∃wex 1779   ∈ wcel 2109  ∀wral 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-ral 3046

This theorem is referenced by: (None)