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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 2138 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∀wal 1361 ∃wex 1502 |
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 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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