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Mirrors > Home > ILE Home > Th. List > celaront | GIF version |
Description: "Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
celaront.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
celaront.min | ⊢ ∀𝑥(𝜒 → 𝜑) |
celaront.e | ⊢ ∃𝑥𝜒 |
Ref | Expression |
---|---|
celaront | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | celaront.maj | . 2 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
2 | celaront.min | . 2 ⊢ ∀𝑥(𝜒 → 𝜑) | |
3 | celaront.e | . 2 ⊢ ∃𝑥𝜒 | |
4 | 1, 2, 3 | barbari 2140 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∀wal 1362 ∃wex 1503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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