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Theorem empty-surprise 45795
 Description: Demonstrate that when using restricted "for all" over a class the expression can be both always true and always false if the class is empty. Those inexperienced with formal notations of classical logic can be surprised with what restricted "for all" does over an empty set. It is important to note that ∀𝑥 ∈ 𝐴𝜑 is simply an abbreviation for ∀𝑥(𝑥 ∈ 𝐴 → 𝜑) (per df-ral 3075). Thus, if 𝐴 is the empty set, this expression is always true regardless of the value of 𝜑 (see alimp-surprise 45793). If you want the expression ∀𝑥 ∈ 𝐴𝜑 to not be vacuously true, you need to ensure that set 𝐴 is inhabited (e.g., ∃𝑥 ∈ 𝐴). (Technical note: You can also assert that 𝐴 ≠ ∅; this is an equivalent claim in classical logic as proven in n0 4247, but in intuitionistic logic the statement 𝐴 ≠ ∅ is a weaker claim than ∃𝑥 ∈ 𝐴.) Some materials on logic (particularly those that discuss "syllogisms") are based on the much older work by Aristotle, but Aristotle expressly excluded empty sets from his system. Aristotle had a specific goal; he was trying to develop a "companion-logic" for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature... This is why he leaves no room for such nonexistent entities in his logic." (Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/aris-log/ 4247). While this made sense for his purposes, it is less flexible than modern (classical) logic which does permit empty sets. If you wish to make claims that require a nonempty set, you must expressly include that requirement, e.g., by stating ∃𝑥𝜑. Examples of proofs that do this include barbari 2690, celaront 2692, and cesaro 2699. For another "surprise" for new users of classical logic, see alimp-surprise 45793 and eximp-surprise 45797. (Contributed by David A. Wheeler, 20-Oct-2018.)
Hypothesis
Ref Expression
empty-surprise.1 ¬ ∃𝑥 𝑥𝐴
Assertion
Ref Expression
empty-surprise 𝑥𝐴 𝜑

Proof of Theorem empty-surprise
StepHypRef Expression
1 empty-surprise.1 . . . 4 ¬ ∃𝑥 𝑥𝐴
21alimp-surprise 45793 . . 3 (∀𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
32simpli 487 . 2 𝑥(𝑥𝐴𝜑)
4 df-ral 3075 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
53, 4mpbir 234 1 𝑥𝐴 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wral 3070 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 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-ral 3075 This theorem is referenced by:  empty-surprise2  45796
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