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Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexmoeu 2601 Existence is equivalent to uniqueness implying existential uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof shortened by BJ, 7-Oct-2022.)
(∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Theoremmoeuex 2602 Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022.)
(∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Theoremmoeu 2603 Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2551 was then proved as dfmo 2615. (Revised by BJ, 30-Sep-2022.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Theoremeubi 2604 Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Theoremeubii 2605 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)

Theoremeubidv 2606* Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremeubid 2607 Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 19-Feb-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

TheoremeubidOLD 2608 Obsolete version of eubid 2607 as of 19-Feb-2023. (Contributed by NM, 9-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Theoremnfeu1 2609 Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT 2610. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥∃!𝑥𝜑

Theoremnfeu1ALT 2610 Alternate proof of nfeu1 2609. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1957 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥∃!𝑥𝜑

Theoremnfeud2 2611 Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfeud 2612 Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2613. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfeu 2613 Bound-variable hypothesis builder for the unique existential quantifier. Note that 𝑥 and 𝑦 need not be disjoint. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝜑       𝑥∃!𝑦𝜑

Theoremdfeu 2614 Rederive df-eu 2587 from the old definition eu6 2592. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof shortened by BJ, 7-Oct-2022.) (Proof modification is discouraged.) Use df-eu 2587 instead. (New usage is discouraged.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

Theoremdfmo 2615* Rederive df-mo 2551 from the old definition moeu 2603. (Contributed by Wolf Lammen, 27-May-2019.) (Proof modification is discouraged.) Use df-mo 2551 instead. (New usage is discouraged.)
(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremeuequ 2616* There exists a unique set equal to a given set. Special case of eueqi 3591 proved using only predicate calculus. The proof needs 𝑦 = 𝑧 be free of 𝑥. This is ensured by having 𝑥 and 𝑦 be distinct. Alternately, a distinctor ¬ ∀𝑥𝑥 = 𝑦 could have been used instead. See eueq 3589 and eueqi 3591 for classes. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) Reduce axiom usage. (Revised by Wolf Lammen, 1-Mar-2023.)
∃!𝑥 𝑥 = 𝑦

TheoremeuequOLD 2617* Obsolete proof of euequ 2616 as of 28-Feb-2023. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∃!𝑥 𝑥 = 𝑦

TheoremmoeuOLD 2618 Obsolete proof of moeu 2603 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

TheoremexmoOLD 2619 Obsolete proof of exmo 2554 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)

TheoremeubidvOLD 2620* Obsolete proof of eubidv 2606 as of 1-Oct-2022. (Contributed by NM, 9-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

TheoremmobidvOLDOLD 2621* Obsolete proof of mobidv 2564 as of 1-Oct-2022. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

Theoremnfmo1OLD 2622 Obsolete proof of nfmo1 2573 as of 1-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥∃*𝑥𝜑

Theoremnfeud2OLD 2623 Obsolete proof of nfeud2 2611 as of 14-Oct-2022. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Theoremnfmod2OLD 2624 Obsolete proof of nfmod2 2574 as of 14-Oct-2022. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

TheoremeubidvOLDOLD 2625* Obsolete version of eubidv 2606 as of 26-Sep-2022. (Contributed by NM, 9-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

TheoremmobidvOLD 2626* Obsolete version of mobidv 2564 as of 7-Oct-2022. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

TheoremeubidOLDOLD 2627 Obsolete proof of eubid 2607 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

TheoremmobidOLDOLD 2628 Obsolete proof of mobid 2565 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))

TheoremeuexOLD 2629 Obsolete proof of euex 2597 as of 7-Oct-2022. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

TheoremdfeuOLD 2630 Obsolete proof of dfeu 2614 as of 7-Oct-2022. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 25-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))

TheoremmoabsOLD 2631 Obsolete proof of moabs 2555 as of 14-Oct-2022. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

TheoremexmoeuOLD 2632 Obsolete proof of exmoeu 2601 as of 7-Oct-2022. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Theoremsb8eulem 2633* Lemma. Factor out the common proof skeleton of sb8euv 2634 and sb8eu 2635. Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.)
𝑦[𝑤 / 𝑥]𝜑    &   (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8euv 2634* Variable substitution in unique existential quantifier. Version of sb8eu 2635 requiring more disjoint variables, but fewer axioms. (Contributed by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8eu 2635 Variable substitution in unique existential quantifier. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2634. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8mo 2636 Variable substitution for the at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑦𝜑       (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Theoremcbvmo 2637 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Theoremcbveu 2638 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

TheoremcbveuALT 2639 Alternative proof of cbveu 2638. Since df-eu 2587 combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

TheoremcbvmoOLD 2640 Obsolete version of cbvmo 2637 as of 4-Jan-2023. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Theoremeu2 2641* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
𝑦𝜑       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Theoremeu1 2642* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) Avoid ax-13 2334. (Revised by Wolf Lammen, 7-Feb-2023.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))

Theoremeu1OLD 2643* Obsolete version of eu1 2642 as of 7-Feb-2023. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝜑       (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))

TheoremeuexALTOLD 2644 Obsolete proof of euex 2597 as of 7-Oct-2022. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥𝜑 → ∃𝑥𝜑)

Theoremeuor 2645 Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2646. (Contributed by NM, 21-Oct-2005.)
𝑥𝜑       ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuorv 2646* Introduce a disjunct into a unique existential quantifier. Version of euor 2645 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

TheoremeuorvOLD 2647* Obsolete version of euorv 2646 as of 14-Jan-2023. (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuor2 2648 Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
(¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Theoremsbmo 2649* Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009.)
([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Theoremeu4 2650* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))

Theoremeuimmo 2651 Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Theoremeuim 2652 Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))

Theoremmoanimlem 2653 Factor out the common proof skeleton of moanimv 2654 and moanim 2655. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.) Factor out common proof lines. (Revised by Wolf Lammen, 8-Feb-2023.)
(𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥(𝜑𝜓)))    &   (∃𝑥(𝜑𝜓) → 𝜑)       (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremmoanimv 2654* Introduction of a conjunct into an at-most-one quantifier. Version of moanim 2655 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce axiom usage . (Revised by Wolf Lammen, 8-Feb-2023.)
(∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremmoanim 2655 Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2654. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
𝑥𝜑       (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremeuan 2656 Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
𝑥𝜑       (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

TheoremmoanimvOLD 2657* Obsolete version of moanimv 2654 as of 8-Feb-2023. (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremmoanmo 2658 Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Theoremmoaneu 2659 Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Theoremeuanv 2660* Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

TheoremeuanvOLD 2661* Obsolete version of euanv 2660 as of 14-Jan-2023. (Contributed by NM, 23-Mar-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Theoremmopick 2662 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Theoremeupick 2663 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Theoremeupicka 2664 Version of eupick 2663 with closed formulas. (Contributed by NM, 6-Sep-2008.)
((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))

Theoremeupickb 2665 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Theoremeupickbi 2666 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
(∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theoremmopick2 2667 "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1915. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))

Theoremmoexex 2668 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
𝑦𝜑       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theoremmoexexv 2669* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theorem2moex 2670 Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
(∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Theorem2euex 2671 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Theorem2eumo 2672 Nested unique existential quantifier and at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Theorem2eu2ex 2673 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)

Theorem2moswap 2674 A condition allowing to swap an existential quantifier and at at-most-one quantifier. (Contributed by NM, 10-Apr-2004.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Theorem2euswap 2675 A condition allowing to swap an existential quanfitier and a unique existential quantifier. (Contributed by NM, 10-Apr-2004.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Theorem2exeu 2676 Double existential uniqueness implies double unique existential quantification. The converse does not hold. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Theorem2mo2 2677* Two ways of expressing "there exists at most one ordered pair 𝑥, 𝑦 such that 𝜑(𝑥, 𝑦) holds. Note that this is not equivalent to ∃*𝑥∃*𝑦𝜑. See also 2mo 2678. This is the analogue of 2eu4 2685 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.)
((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2mo 2678* Two ways of expressing "there exists at most one ordered pair 𝑥, 𝑦 such that 𝜑(𝑥, 𝑦) holds. See also 2mo2 2677. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
(∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2mos 2679* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2eu1 2680 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu1OLD 2681 Obsolete version of 2eu1 2680 as of 23-Apr-2023. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu2 2682 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))

Theorem2eu3 2683 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.)
(∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu3OLD 2684 Obsolete version of 2eu3 2683 as of 23-Apr-2023. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu4 2685* This theorem provides us with a definition of double existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"). Naively one might think (incorrectly) that it could be defined by ∃!𝑥∃!𝑦𝜑. See 2eu1 2680 for a condition under which the naive definition holds and 2exeu 2676 for a one-way implication. See 2eu5 2686 and 2eu8 2689 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu5 2686* An alternate definition of double existential uniqueness (see 2eu4 2685). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦". (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published (∃* means "exists at most one"). (Contributed by NM, 26-Oct-2003.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu6 2687* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2eu7 2688 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Theorem2eu8 2689 Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2688. (Contributed by NM, 20-Feb-2005.)
(∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Theoremeuae 2690* Two ways to express "exactly one thing exists". To paraphrase the statement and explain the label: there Exists a Unique thing if and only if for All 𝑥, 𝑥 Equals some given (and disjoint) 𝑦. Both sides are false in set theory, see theorems neutru 32998 and dtru 5084. (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies. (Revised by Wolf Lammen, 2-Mar-2023.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

TheoremeuaeOLD 2691* Obsolete version of euae 2690 as of 2-Mar-2023. (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremexists1 2692* Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see theorem dtru 5084. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.)
(∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremexists2 2693 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 4-Mar-2023.)
((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

Theoremexists2OLD 2694 Obsolete version of exists2 2693 as of 4-Mar-2023. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((∃𝑥𝜑 ∧ ∃𝑥 ¬ 𝜑) → ¬ ∃!𝑥 𝑥 = 𝑥)

1.7  Other axiomatizations related to classical predicate calculus

1.7.1  Aristotelian logic: Assertic syllogisms

Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems.

In antiquity Aristotelian logic and Stoic logic (see mptnan 1812) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe. This section models this system (including later refinements). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order.

"There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation.

We instead translate each Aristotelian syllogism into an inference rule, and each rule is defined using standard predicate logic notation and predicates. The predicates are represented by wff variables that may depend on the quantified variable 𝑥. Our translation is essentially identical to the one used in Rini page 18, Table 2 "Non-Modal Syllogisms in Lower Predicate Calculus (LPC)", which uses standard predicate logic with predicates. Rini states, "the crucial point is that we capture the meaning Aristotle intends, and the method by which we represent that meaning is less important". There are two differences: we make the existence criteria explicit, and we use 𝜑, 𝜓, and 𝜒 in the order they appear (a common Metamath convention). Patzig also uses standard predicate logic notation and predicates (though he interprets them as conditional propositions, not as inference rules); see Gunther Patzig, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the approach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26).

Expressions of the form "no 𝜑 is 𝜓 " are consistently translated as 𝑥(𝜑 → ¬ 𝜓). These can also be expressed as ¬ ∃𝑥(𝜑𝜓), per alinexa 1888. We translate "all 𝜑 is 𝜓 " to 𝑥(𝜑𝜓), "some 𝜑 is 𝜓 " to 𝑥(𝜑𝜓), and "some 𝜑 is not 𝜓 " to 𝑥(𝜑 ∧ ¬ 𝜓). It is traditional to use the singular form "is", not the plural form "are", in the generic expressions. By convention the major premise is listed first.

In traditional Aristotelian syllogisms the predicates have a restricted form ("x is a ..."); those predicates could be modeled in modern notation by more specific constructs such as 𝑥 = 𝐴, 𝑥𝐴, or 𝑥𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥𝐴 instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion () and membership (); this distinction was not directly made in Aristotle's work.

There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such nonexistent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2701, celaront 2703, cesaro 2712, camestros 2714, felapton 2724, darapti 2722, calemos 2731, fesapo 2733, and bamalip 2735.

These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically, Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here.

Aristotelian logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus.

The following twenty-four syllogisms (from barbara 2695 to bamalip 2735) are all proven from { ax-mp 5, ax-1 6, ax-2 7, ax-3 8, ax-gen 1839, ax-4 1853 }, which corresponds in the usual translation to modal logic (a universal (resp. existential) quantifier maps to necessity (resp. possibility)) to the weakest normal modal logic (K). Some proofs could be shortened by using additionally spi 2168 (inference form of sp 2167, which corresponds to the axiom (T) of modal logic), as demonstrated by dariiALT 2698, barbariALT 2702, festinoALT 2709, barocoALT 2711, daraptiALT 2723.

Theorembarbara 2695 "Barbara", one of the fundamental syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore all 𝜒 is 𝜓. In Aristotelian notation, AAA-1: MaP and SaM therefore SaP. For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as 𝑥(𝑥𝐻𝑥𝑀) (all men are mortal) and 𝑥(𝑥 = 𝑆𝑥𝐻) (Socrates is a man) therefore 𝑥(𝑥 = 𝑆𝑥𝑀) (Socrates is mortal). Russell and Whitehead note that "the syllogism in Barbara is derived from [syl 17]" (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1939. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

Theoremcelarent 2696 "Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. Instance of barbara 2695. In Aristotelian notation, EAE-1: MeP and SaM therefore SeP. For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒 → ¬ 𝜓)

Theoremdarii 2697 "Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-1: MaP and SiM therefore SiP. For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. See dariiALT 2698 for a shorter proof requiring more axioms. (Contributed by David A. Wheeler, 24-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

TheoremdariiALT 2698 Alternate proof of darii 2697, shorter but using more axioms. This shows how the use of spi 2168 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2168 is the inference associated with sp 2167, which corresponds to the axiom (T) of modal logic. (Contributed by David A. Wheeler, 27-Aug-2016.) Added precisions on axiom usage. (Revised by BJ, 27-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

Theoremferio 2699 "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. Instance of darii 2697. In Aristotelian notation, EIO-1: MeP and SiM therefore SoP. For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒 ∧ ¬ 𝜓)

Theorembarbarilem 2700 Lemma for barbari 2701 and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022.)
𝑥𝜑    &   𝑥(𝜑𝜓)       𝑥(𝜑𝜓)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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