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Mirrors > Home > MPE Home > Th. List > celaront | Structured version Visualization version 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 2555 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
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-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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