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Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembarbari 2701 "Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-1: MaP and SaM therefore SiP. For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒𝜓)
 
TheorembarbariALT 2702 Alternate proof of barbari 2701, shorter but using more axioms. See comment of dariiALT 2698. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒𝜓)
 
Theoremcelaront 2703 "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 2701. 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.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜓)
 
Theoremcesare 2704 "Cesare", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent 2696. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 → ¬ 𝜑)
 
TheoremcesareOLD 2705 Obsolete proof of cesare 2704 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 → ¬ 𝜑)
 
Theoremcamestres 2706 "Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)       𝑥(𝜒 → ¬ 𝜑)
 
TheoremcamestresOLD 2707 Obsolete proof of camestres 2706 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)       𝑥(𝜒 → ¬ 𝜑)
 
Theoremfestino 2708 "Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremfestinoALT 2709 Alternate proof of festino 2708, shorter but using more axioms. See comment of dariiALT 2698. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theorembaroco 2710 "Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 ∧ ¬ 𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheorembarocoALT 2711 Alternate proof of festino 2708, shorter but using more axioms. See comment of dariiALT 2698. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 ∧ ¬ 𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theoremcesaro 2712 "Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremcesaroOLD 2713 Obsolete proof of cesaro 2712 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theoremcamestros 2714 "Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, AEO-2: PaM and SeM therefore SoP. For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremcamestrosOLD 2715 Obsolete proof of camestros 2714 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theoremdatisi 2716 "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)
 
TheoremdatisiOLD 2717 Obsolete proof of datisi 2716 as of 16-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)
 
Theoremdisamis 2718 "Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)
 
TheoremdisamisOLD 2719 Obsolete proof of disamis 2718 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)
 
Theoremferison 2720 "Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of datisi 2716. In Aristotelian notation, EIO-3: MeP and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒 ∧ ¬ 𝜓)
 
Theorembocardo 2721 "Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of disamis 2718. In Aristotelian notation, OAO-3: MoP and MaS therefore SoP. For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑 ∧ ¬ 𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒 ∧ ¬ 𝜓)
 
Theoremdarapti 2722 "Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-3: MaP and MaS therefore SiP. For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒𝜓)
 
TheoremdaraptiALT 2723 Alternate proof of darapti 2722, shorter but using more axioms. See comment of dariiALT 2698. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒𝜓)
 
Theoremfelapton 2724 "Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. Instance of darapti 2722. In Aristotelian notation, EAO-3: MeP and MaS therefore SoP. For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒 ∧ ¬ 𝜓)
 
Theoremcalemes 2725 "Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)       𝑥(𝜒 → ¬ 𝜑)
 
TheoremcalemesOLD 2726 Obsolete proof of calemes 2725 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)       𝑥(𝜒 → ¬ 𝜑)
 
Theoremdimatis 2727 "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. In Aristotelian notation, IAI-4: PiM and MaS therefore SiP. For example, "Some pets are rabbits", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2697 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒𝜑)
 
TheoremdimatisOLD 2728 Obsolete proof of dimatis 2727 as of 16-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒𝜑)
 
Theoremfresison 2729 "Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, EIO-4: PeM and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremfresisonOLD 2730 Obsolete proof of fresison 2729 as of 16-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theoremcalemos 2731 "Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, AEO-4: PaM and MeS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremcalemosOLD 2732 Obsolete proof of calemos 2731 as of 16-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theoremfesapo 2733 "Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EAO-4: PeM and MaS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜓       𝑥(𝜒 ∧ ¬ 𝜑)
 
TheoremfesapoOLD 2734 Obsolete proof of fesapo 2733 as of 27-Sep-2022. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜓       𝑥(𝜒 ∧ ¬ 𝜑)
 
Theorembamalip 2735 "Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. In Aristotelian notation, AAI-4: PaM and MaS therefore SiP. Very similar to barbari 2701. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜑       𝑥(𝜒𝜑)
 
TheorembamalipOLD 2736 Obsolete proof of bamalip 2735 as of 16-Sep-2022. (Contributed by David A. Wheeler, 28-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜑       𝑥(𝜒𝜑)
 
1.7.2  Intuitionistic logic

Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 881 or peirce 194. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 837 and df-ex 1824 which are not valid in intuitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm.

The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1853, ax-11 2150, ax-gen 1839, ax-7 2055, ax-12 2163, ax-8 2109, ax-9 2116, and ax-5 1953.

In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.

 
Theoremaxia1 2737 Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜑)
 
Theoremaxia2 2738 Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜓)
 
Theoremaxia3 2739 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜑𝜓)))
 
Theoremaxin1 2740 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)
 
Theoremaxin2 2741 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
𝜑 → (𝜑𝜓))
 
Theoremaxio 2742 Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremaxi4 2743 Specialization (intuitionistic logic axiom ax-4). This is just sp 2167 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremaxi5r 2744 Converse of axc4 2296 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))
 
Theoremaxial 2745 The setvar 𝑥 is not free in 𝑥𝜑 (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremaxie1 2746 The setvar 𝑥 is not free in 𝑥𝜑 (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremaxie2 2747 A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
(∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 
Theoremaxi9 2748 Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 2021 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦
 
Theoremaxi10 2749 Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2392 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxi12 2750 Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2346 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.) Avoid ax-11 2150. (Revised by Wolf Lammen, 24-Apr-2023.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
Theoremaxi12OLD 2751 Obsolete version of axi12 2750 as of 24-Apr-2023. (Contributed by Jim Kingdon, 31-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
Theoremaxbnd 2752 Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2750 are fairly straightforward consequences of axc9 2346. But in intuitionistic logic, it is not easy to add the extra 𝑥 to axi12 2750 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof shortened by Wolf Lammen, 24-Apr-2023.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
TheoremaxbndOLD 2753 Obsolete version of axbnd 2752 as of 24-Apr-2023. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
 
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY

Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets". A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects.

A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3651), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here.

Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties". (The axiom of Extensionality can also be restated this way as shown by axext2 2755.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems.

 
2.1  ZF Set Theory - start with the Axiom of Extensionality
 
2.1.1  Introduce the Axiom of Extensionality
 
Axiomax-ext 2754* Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461.

Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤𝑥𝑤𝑦) → (𝑥𝑧𝑦𝑧)), and equality 𝑥 = 𝑦 is defined as 𝑤(𝑤𝑥𝑤𝑦). All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8.

To use the above "equality-free" version of Extensionality with Metamath's predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully.

General remarks: Our set theory axioms are presented using defined connectives (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , ¬, , =, and . It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so.

It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable 𝑥 in ax-ext 2754 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both 𝑥 and 𝑧. This is in contrast to typical textbook presentations that present actual axioms (except for Replacement ax-rep 5006, which involves a wff metavariable). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2754 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)

(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theoremaxext2 2755* The Axiom of Extensionality (ax-ext 2754) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theoremaxext3 2756* A generalization of the Axiom of Extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Remove dependencies on ax-10 2135, ax-12 2163, ax-13 2334. (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theoremaxext3ALT 2757* Alternate proof of axext3 2756, shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theoremaxext4 2758* A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2754 and df-cleq 2770. (Contributed by NM, 14-Nov-2008.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
 
Theoremaxextmo 2759* There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)
𝑥𝜑       ∃*𝑥𝑦(𝑦𝑥𝜑)
 
Theorembm1.1OLD 2760* Obsolete version of axextmo 2759 as of 17-Sep-2022. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) (Proof modification is discouraged.) Use axextmo 2759 instead. (New usage is discouraged.)
𝑥𝜑       (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))
 
Theoremnulmo 2761* There exists at most one empty set. With either axnul 5024 or axnulALT 5023 or ax-nul 5025, this proves that there exists a unique empty set. In practice, once the language of classes is available, we use the stronger characterization among classes eq0 4157. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof shortened by Wolf Lammen, 26-Apr-2023.)
∃*𝑥𝑦 ¬ 𝑦𝑥
 
TheoremnulmoOLD 2762* Obsolete version of nulmo 2761 as of 26-Apr-2023. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
∃*𝑥𝑦 ¬ 𝑦𝑥
 
2.1.2  Class abstractions (a.k.a. class builders)
 
Syntaxcab 2763 Introduce the class builder or class abstraction notation ("the class of sets 𝑥 such that 𝜑 is true"). Our class variables 𝐴, 𝐵, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 4142). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2772, justifying the assignment of setvar variables to class variables via the use of cv 1600.
class {𝑥𝜑}
 
Definitiondf-clab 2764 Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. 𝑥 and 𝑦 need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, 𝜑 will have 𝑦 as a free variable, and "{𝑦𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true." We do not define {𝑦𝜑} in isolation but only as part of an expression that extends or "overloads" the relationship.

This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 2107, which extends or "overloads" the wel 2108 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2770 and df-clel 2774, we introduce a new kind of variable (class variable) that can be substituted with expressions such as {𝑦𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1600 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2772 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2892 for a quick overview).

Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is done with theorems such as vtoclg 3467 which is used, for example, to convert elirrv 8790 to elirr 8791.

This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction {𝑦𝜑} a "class term".

While the three class definitions df-clab 2764, df-cleq 2770, and df-clel 2774 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
 
Theoremabid 2765 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.)
(𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
 
Theoremhbab1 2766* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
 
Theoremnfsab1 2767* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦 ∈ {𝑥𝜑}
 
Theoremhbab 2768* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
(𝜑 → ∀𝑥𝜑)       (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
 
Theoremnfsab 2769* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥 𝑧 ∈ {𝑦𝜑}
 
Definitiondf-cleq 2770* Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs.

This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce 𝑦 = 𝑧 ↔ ∀𝑥(𝑥𝑦𝑥𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2758). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

See also comments under df-clab 2764, df-clel 2774, and abeq2 2892.

In the form of dfcleq 2771, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

While the three class definitions df-clab 2764, df-cleq 2770, and df-clel 2774 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 15-Sep-1993.)

(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdfcleq 2771* The same as df-cleq 2770 with the hypothesis removed using the Axiom of Extensionality ax-ext 2754. (Contributed by NM, 15-Sep-1993.) Revised to make use of axext3 2756 instead of ax-ext 2754, so that ax-9 2116 will appear in lists of axioms used by a proof, since df-cleq 2770 implies ax-9 2116 by theorem bj-ax9 33461. We may revisit this in the future. (Revised by NM, 28-Oct-2021.) (Proof modification is discouraged.)
(𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremcvjust 2772* Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1600, which allows us to substitute a setvar variable for a class variable. See also cab 2763 and df-clab 2764. Note that this is not a rigorous justification, because cv 1600 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." See abid1 2911 for the version of cvjust 2772 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2334. (Revised by Wolf Lammen, 4-May-2023.)
𝑥 = {𝑦𝑦𝑥}
 
TheoremcvjustOLD 2773* Obsolete version of cvjust 2772 as of 4-May-2023. (Contributed by NM, 7-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = {𝑦𝑦𝑥}
 
Definitiondf-clel 2774* Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 2770 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2770 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2115), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2764. Alternate definitions of 𝐴𝐵 (but that require either 𝐴 or 𝐵 to be a set) are shown by clel2 3543, clel3 3545, and clel4 3546.

This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms.

While the three class definitions df-clab 2764, df-cleq 2770, and df-clel 2774 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)

(𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
 
Theoremeqriv 2775* Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.)
(𝑥𝐴𝑥𝐵)       𝐴 = 𝐵
 
Theoremeqrdv 2776* Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremeqrdav 2777* Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
((𝜑𝑥𝐴) → 𝑥𝐶)    &   ((𝜑𝑥𝐵) → 𝑥𝐶)    &   ((𝜑𝑥𝐶) → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremeqid 2778 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 253. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.)

𝐴 = 𝐴
 
Theoremeqidd 2779 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
(𝜑𝐴 = 𝐴)
 
Theoremeqeq1d 2780 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))
 
Theoremeqeq1dALT 2781 Shorter proof of eqeq1d 2780 based on more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐶))
 
Theoremeqeq1 2782 Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵 → (𝐴 = 𝐶𝐵 = 𝐶))
 
Theoremeqeq1i 2783 Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.)
𝐴 = 𝐵       (𝐴 = 𝐶𝐵 = 𝐶)
 
Theoremeqcomd 2784 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2785. (Revised by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑𝐵 = 𝐴)
 
Theoremeqcom 2785 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵𝐵 = 𝐴)
 
Theoremeqcoms 2786 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.)
(𝐴 = 𝐵𝜑)       (𝐵 = 𝐴𝜑)
 
Theoremeqcomi 2787 Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵       𝐵 = 𝐴
 
Theoremeqeq2d 2788 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2789. (Revised by Wolf Lammen, 19-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶 = 𝐴𝐶 = 𝐵))
 
Theoremeqeq2 2789 Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
(𝐴 = 𝐵 → (𝐶 = 𝐴𝐶 = 𝐵))
 
Theoremeqeq2i 2790 Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.)
𝐴 = 𝐵       (𝐶 = 𝐴𝐶 = 𝐵)
 
Theoremeqeq12 2791 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeq12i 2792 A useful inference for substituting definitions into an equality. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴 = 𝐶𝐵 = 𝐷)
 
Theoremeqeq12d 2793 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan12d 2794 A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2795. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan12dALT 2795 Alternate proof of eqeqan12d 2794. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan12rd 2796 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan1dOLD 2797 Implication of introducing a new equality. Obsolete as of 14-Feb-2023. Use eqeqan12d 2794 instead. (Contributed by Peter Mazsa, 17-Apr-2019.) (Proof shortened by AV, 10-Feb-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐴 = 𝐵)       ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqeqan1dOLDOLD 2798 Obsolete proof of eqeqan1dOLD 2797 as of 10-Feb-2023. (Contributed by Peter Mazsa, 17-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 = 𝐵)       ((𝜑𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremeqtr 2799 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
 
Theoremeqtr2 2800 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝐴 = 𝐵𝐴 = 𝐶) → 𝐵 = 𝐶)
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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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