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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. Instance of barbari 2754. 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 2754. (Contributed by David A. Wheeler, 27-Aug-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 2754 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: (None) |
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