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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 42337. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1510); 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 42344. (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 42337 | 1 ⊢ ∀𝑥 ∈ 𝐴 ⊥ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊥wfal 1480 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-ral 2901 |
This theorem is referenced by: (None) |
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