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Mirrors > Home > MPE Home > Th. List > Mathboxes > empty-surprise2 | Structured version Visualization version GIF version |
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 44811. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1556); 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 44818. (Contributed by David A. Wheeler, 20-Oct-2018.) |
Ref | Expression |
---|---|
empty-surprise2.1 | ⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 |
Ref | Expression |
---|---|
empty-surprise2 | ⊢ ∀𝑥 ∈ 𝐴 ⊥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | empty-surprise2.1 | . 2 ⊢ ¬ ∃𝑥 𝑥 ∈ 𝐴 | |
2 | 1 | empty-surprise 44811 | 1 ⊢ ∀𝑥 ∈ 𝐴 ⊥ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊥wfal 1540 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-ral 3140 |
This theorem is referenced by: (None) |
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