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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 42528. Here we prove an extreme example: we "prove" that false is true. Of course, we actually do no such thing (see notfal 1519); 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 42535. (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 42528 | 1 ⊢ ∀𝑥 ∈ 𝐴 ⊥ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ⊥wfal 1488 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-ral 2917 |
This theorem is referenced by: (None) |
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