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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 44890. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1565); 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 44897. (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 44890 | 1 ⊢ ∀𝑥 ∈ 𝐴 ⊥ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊥wfal 1549 ∃wex 1780 ∈ wcel 2114 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-ral 3145 |
This theorem is referenced by: (None) |
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