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Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisbasisrelowl 33801 The set of all closed-below, open-above intervals of reals form a basis. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       𝐼 ∈ TopBases

Theoremicoreunrn 33802 The union of all closed-below, open-above intervals of reals is the set of reals. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ℝ = 𝐼

Theoremistoprelowl 33803 The set of all closed-below, open-above intervals of reals generate a topology on the reals. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       (topGen‘𝐼) ∈ (TopOn‘ℝ)

Theoremicoreelrn 33804* A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴𝑧𝑧 < 𝐵)} ∈ 𝐼)

Theoremiooelexlt 33805* An element of an open interval is not its smallest element. (Contributed by ML, 2-Aug-2020.)
(𝑋 ∈ (𝐴(,)𝐵) → ∃𝑦 ∈ (𝐴(,)𝐵)𝑦 < 𝑋)

Theoremrelowlssretop 33806 The lower limit topology on the reals is finer than the standard topology. (Contributed by ML, 1-Aug-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       (topGen‘ran (,)) ⊆ (topGen‘𝐼)

Theoremrelowlpssretop 33807 The lower limit topology on the reals is strictly finer than the standard topology. (Contributed by ML, 2-Aug-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       (topGen‘ran (,)) ⊊ (topGen‘𝐼)

Theoremsucneqond 33808 Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)
(𝜑𝑋 = suc 𝑌)    &   (𝜑𝑌 ∈ On)       (𝜑𝑋𝑌)

Theoremsucneqoni 33809 Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)
𝑋 = suc 𝑌    &   𝑌 ∈ On       𝑋𝑌

Theoremonsucuni3 33810 If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.)
((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc 𝐵)

Theorem1oequni2o 33811 The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.)
1o = 2o

Theoremrdgsucuni 33812 If an ordinal number has a predecessor, the value of the recursive definition generator at that number in terms of its predecessor. (Contributed by ML, 17-Oct-2020.)
((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (rec(𝐹, 𝐼)‘𝐵) = (𝐹‘(rec(𝐹, 𝐼)‘ 𝐵)))

Theoremrdgeqoa 33813 If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)
((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))

Theoremelxp8 33814 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7480. (Contributed by ML, 19-Oct-2020.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))

Syntaxcfinxp 33815 Extend the definition of a class to include Cartesian exponentiation.
class (𝑈↑↑𝑁)

Definitiondf-finxp 33816* Define Cartesian exponentiation on a class.

Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 8195 or df-map 8142 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2o), then df-br 4887 can be used on it, and df-fv 6143 can also be used, and so on.

It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +o 𝑁)).

This definition is technical. Use finxp1o 33824 and finxpsuc 33830 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.)

(𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

Theoremdffinxpf 33817* This theorem is the same as the definition df-finxp 33816, except that the large function is replaced by a class variable for brevity. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}

Theoremfinxpeq1 33818 Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
(𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Theoremfinxpeq2 33819 Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
(𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Theoremcsbfinxpg 33820* Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))

Theoremfinxpreclem1 33821* Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
(𝑋𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))

Theoremfinxpreclem2 33822* Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
((𝑋 ∈ V ∧ ¬ 𝑋𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))

Theoremfinxp0 33823 The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑∅) = ∅

Theoremfinxp1o 33824 The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)
(𝑈↑↑1o) = 𝑈

Theoremfinxpreclem3 33825* Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Theoremfinxpreclem4 33826* Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))

Theoremfinxpreclem5 33827* Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑛 ∈ ω ∧ 1o𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Theoremfinxpreclem6 33828* Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Theoremfinxpsuclem 33829* Lemma for finxpsuc 33830. (Contributed by ML, 24-Oct-2020.)
𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))       ((𝑁 ∈ ω ∧ 1o𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theoremfinxpsuc 33830 The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.)
((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theoremfinxp2o 33831 The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)
(𝑈↑↑2o) = (𝑈 × 𝑈)

Theoremfinxp3o 33832 The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈)

Theoremfinxpnom 33833 Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)
𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅)

Theoremfinxp00 33834 Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)
(∅↑↑𝑁) = ∅

20.16.1  Cantor normal form up to epsilon 0

Definition of the Cantor normal form for ordinals up to epsilon 0, without using the axiom of infinity.

Theoremcnfin0 33835 The empty set is an ordinal in Cantor normal form. (Contributed by ML, 24-Jun-2022.)
𝐼 = {⟨∅, 1o⟩}    &    + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +o (𝑧𝑛))))    &   (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))    &   (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))    &   (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1o⟩} ∧ 𝑦 = {⟨𝑏, 1o⟩})))    &   𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}    &    < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)    &   𝐶 = dom <       ∅ ∈ 𝐶

Theoremcnfinltrel 33836* Less than for the Cantor normal form is a relation. (Contributed by ML, 24-Jun-2022.)
𝐼 = {⟨∅, 1o⟩}    &    + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +o (𝑧𝑛))))    &   (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))    &   (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))    &   (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1o⟩} ∧ 𝑦 = {⟨𝑏, 1o⟩})))    &   𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}    &    < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)    &   𝐶 = dom <       Rel <

20.17  Mathbox for Wolf Lammen

Theoremwl-section-prop 33837 Intuitionistic logic is now developed separately, so we need not first focus on intuitionally valid axioms ax-1 6 and ax-2 7 any longer.

Alternatively, I start from Jan Lukasiewicz's axiom system here, i.e. ax-mp 5, ax-luk1 33838, ax-luk2 33839 and ax-luk3 33840. I rather copy this system than use luk-1 1699 to luk-3 1701, since the latter are theorems, while we need axioms here.

(Contributed by Wolf Lammen, 23-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜑       𝜑

Axiomax-luk1 33838 1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1699 and imim1 83, but introduced as an axiom. It focuses on a basic property of a valid implication, namely that the consequent has to be true whenever the antecedent is. So if 𝜑 and 𝜓 are somehow parametrized expressions, then 𝜑𝜓 states that 𝜑 strengthen 𝜓, in that 𝜑 holds only for a (often proper) subset of those parameters making 𝜓 true. We easily accept, that when 𝜓 is stronger than 𝜒 and, at the same time 𝜑 is stronger than 𝜓, then 𝜑 must be stronger than 𝜒. This transitivity is expressed in this axiom.

A particular result of this strengthening property comes into play if the antecedent holds unconditionally. Then the consequent must hold unconditionally as well. This specialization is the foundational idea behind logical conclusion. Such conclusion is best expressed in so-called immediate versions of this axiom like imim1i 63 or syl 17. Note that these forms are weaker replacements (i.e. just frequent specialization) of the closed form presented here, hence a mere convenience.

We can identify in this axiom up to three antecedents, followed by a consequent. The number of antecedents is not really fixed; the fewer we are willing to "see", the more complex the consequent grows. On the other side, since 𝜒 is a variable capable of assuming an implication itself, we might find even more antecedents after some substitution of 𝜒. This shows that the ideas of antecedent and consequent in expressions like this depends on, and can adapt to, our current interpretation of the whole expression.

In this axiom, up to two antecedents happen to be of complex nature themselves, i.e. are an embedded implication. Logically, this axiom is a compact notion of simpler expressions, which I will later coin implication chains. Herein all antecedents and the consequent appear as simple variables, or their negation. Any propositional expression is equivalent to a set of such chains. This axiom, for example, is dissected into following chains, from which it can be recovered losslessly:

(𝜓 → (𝜒 → (𝜑𝜒))); 𝜑 → (𝜒 → (𝜑𝜒))); (𝜓 → (¬ 𝜓 → (𝜑𝜒))); 𝜑 → (¬ 𝜓 → (𝜑𝜒))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Axiomax-luk2 33839 2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1700 or pm2.18 125, but introduced as an axiom. The core idea behind this axiom is, that if something can be implied from both an antecedent, and separately from its negation, then the antecedent is irrelevant to the consequent, and can safely be dropped. This is perhaps better seen from the following slightly extended version (related to pm2.65 185):

((𝜑𝜑) → ((¬ 𝜑𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)

((¬ 𝜑𝜑) → 𝜑)

Axiomax-luk3 33840 3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1701 and pm2.24 122, but introduced as an axiom. One might think that the similar pm2.21 121 𝜑 → (𝜑𝜓)) is a valid replacement for this axiom. But this is not true, ax-3 8 is not derivable from this modification. This can be shown by designing carefully operators ¬ and on a finite set of primitive statements. In propositional logic such statements are and , but we can assume more and other primitives in our universe of statements. So we denote our primitive statements as phi0 , phi1 and phi2. The actual meaning of the statements are not important in this context, it rather counts how they behave under our operations ¬ and , and which of them we assume to hold unconditionally (phi1, phi2). For our disproving model, I give that information in tabular form below. The interested reader may check per hand, that all possible interpretations of ax-mp 5, ax-luk1 33838, ax-luk2 33839 and pm2.21 121 result in phi1 or phi2, meaning they always hold. But for wl-luk-ax3 33852 we can find a counter example resulting in phi0, not a statement always true. The verification of a particular set of axioms in a given model is tedious and error prone, so I wrote a computer program, first checking this for me, and second, hunting for a counter example. Here is the result, after 9165 fruitlessly computer generated models:

ax-3 fails for phi2, phi2
number of statements: 3
always true phi1 phi2

Negation is defined as
----------------------------------------------------------------------
 -. phi0 -. phi1 -. phi2 phi1 phi0 phi1

Implication is defined as
----------------------------------------------------------------------
 p->q q: phi0 q: phi1 q: phi2 p: phi0 phi1 phi1 phi1 p: phi1 phi0 phi1 phi1 p: phi2 phi0 phi0 phi0

(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

20.17.1  1. Bootstrapping

Theoremwl-section-boot 33841 In this section, I provide the first steps needed for convenient proving. The presented theorems follow no common concept other than being useful in themselves, and apt to rederive ax-1 6, ax-2 7 and ax-3 8. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝜑

Theoremwl-luk-imim1i 33842 Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 63 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.)
(𝜑𝜓)       ((𝜓𝜒) → (𝜑𝜒))

Theoremwl-luk-syl 33843 An inference version of the transitive laws for implication luk-1 1699. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremwl-luk-syl5 33844 A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 34 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       (𝜒 → (𝜑𝜃))

Theoremwl-luk-pm2.18d 33845 Deduction based on reductio ad absurdum. Copy of pm2.18d 127 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theoremwl-luk-con4i 33846 Inference rule. Copy of con4i 114 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theoremwl-luk-pm2.24i 33847 Inference rule. Copy of pm2.24i 148 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝜑𝜓)

Theoremwl-luk-a1i 33848 Inference rule. Copy of a1i 11 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       (𝜓𝜑)

Theoremwl-luk-mpi 33849 A nested modus ponens inference. Copy of mpi 20 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremwl-luk-imim2i 33850 Inference adding common antecedents in an implication. Copy of imim2i 16 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremwl-luk-syl6 33851 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 35 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑 → (𝜓𝜃))

Theoremwl-luk-ax3 33852 ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((¬ 𝜑 → ¬ 𝜓) → (𝜓𝜑))

Theoremwl-luk-ax1 33853 ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremwl-luk-pm2.27 33854 This theorem, called "Assertion", can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 42 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))

Theoremwl-luk-com12 33855 Inference that swaps (commutes) antecedents in an implication. Copy of com12 32 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜓 → (𝜑𝜒))

Theoremwl-luk-pm2.21 33856 From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 121 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))

Theoremwl-luk-con1i 33857 A contraposition inference. Copy of con1i 147 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜓)       𝜓𝜑)

Theoremwl-luk-ja 33858 Inference joining the antecedents of two premises. Copy of ja 175 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-luk-imim2 33859 A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremwl-luk-a1d 33860 Deduction introducing an embedded antecedent. Copy of imim2 58 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremwl-luk-ax2 33861 ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremwl-luk-id 33862 Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. Copy of id 22 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜑)

Theoremwl-luk-notnotr 33863 Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true; in intuitionistic logic, when this is true for some 𝜑, then 𝜑 is stable. Copy of notnotr 128 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremwl-luk-pm2.04 33864 Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Copy of pm2.04 90 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

20.17.2  Implication chains

Theoremwl-section-impchain 33865 An implication like (𝜓𝜑) with one antecedent can easily be extended by prepending more and more antecedents, as in (𝜒 → (𝜓𝜑)) or (𝜃 → (𝜒 → (𝜓𝜑))). I call these expressions implication chains, and the number of antecedents (number of nodes minus one) denotes their length. A given length often marks just a required minimum value, since the consequent 𝜑 itself may represent an implication, or even an implication chain, such hiding part of the whole chain. As an extension, it is useful to consider a single variable 𝜑 as a degenerate implication chain of length zero.

Implication chains play a particular role in logic, as all propositional expressions turn out to be convertible to one or more implication chains, their nodes as simple as a variable, or its negation.

So there is good reason to focus on implication chains as a sort of normalized expressions, and build some general theorems around them, with proofs using recursive patterns. This allows for theorems referring to longer and longer implication chains in an automated way.

The theorem names in this section contain the text fragment 'impchain' to point out their relevance to implication chains, followed by a number indicating the (minimal) length of the longest chain involved. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜑       𝜑

Theoremwl-impchain-mp-x 33866 This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, the theorems ax-mp 5, syl 17, syl6 35, syl8 76 form the beginning of this series. These theorems are replicated here, but with proofs that aim at a recursive scheme, allowing to base a proof on that of the previous one in the series. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-mp-0 33867 This theorem is the start of a proof recursion scheme where we replace the consequent of an implication chain. The number '0' in the theorem name indicates that the modified chain has no antecedents.

This theorem is in fact a copy of ax-mp 5, and is repeated here to emphasize the recursion using similar theorem names. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

𝜓    &   (𝜓𝜑)       𝜑

Theoremwl-impchain-mp-1 33868 This theorem is in fact a copy of wl-luk-syl 33843, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜒𝜓)    &   (𝜓𝜑)       (𝜒𝜑)

Theoremwl-impchain-mp-2 33869 This theorem is in fact a copy of wl-luk-syl6 33851, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒𝜓))    &   (𝜓𝜑)       (𝜃 → (𝜒𝜑))

Theoremwl-impchain-com-1.x 33870 It is often convenient to have the antecedent under focus in first position, so we can apply immediate theorem forms (as opposed to deduction, tautology form). This series of theorems swaps the first with the last antecedent in an implication chain. This kind of swapping is self-inverse, whence we prefer it over, say, rotating theorems. A consequent can hide a tail of a longer chain, so theorems of this series appear as swapping a pair of antecedents with fixed offsets. This form of swapping antecedents is flexible enough to allow for any permutation of antecedents in an implication chain.

The first elements of this series correspond to com12 32, com13 88, com14 96 and com15 101 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 33866 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-1.1 33871 A degenerate form of antecedent swapping. The number '1' in the theorem name indicates that it handles a chain of length 1.

Since there is just one antecedent in the chain, there is nothing to swap. Nondegenerated forms begin with wl-impchain-com-1.2 33872, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜓𝜑)       (𝜓𝜑)

Theoremwl-impchain-com-1.2 33872 This theorem is in fact a copy of wl-luk-com12 33855, and repeated here to demonstrate a simple proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified.

See wl-impchain-com-1.x 33870 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜒 → (𝜓𝜑))       (𝜓 → (𝜒𝜑))

Theoremwl-impchain-com-1.3 33873 This theorem is in fact a copy of com13 88, and repeated here to demonstrate a simple proof scheme. The number '3' in the theorem name indicates that a chain of length 3 is modified.

See wl-impchain-com-1.x 33870 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜒 → (𝜃𝜑)))

Theoremwl-impchain-com-1.4 33874 This theorem is in fact a copy of com14 96, and repeated here to demonstrate a simple proof scheme. The number '4' in the theorem name indicates that a chain of length 4 is modified.

See wl-impchain-com-1.x 33870 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜓 → (𝜃 → (𝜒 → (𝜂𝜑))))

Theoremwl-impchain-com-n.m 33875 This series of theorems allow swapping any two antecedents in an implication chain. The theorem names follow a pattern wl-impchain-com-n.m with integral numbers n < m, that swaps the m-th antecedent with n-th one in an implication chain. It is sufficient to restrict the length of the chain to m, too, since the consequent can be assumed to be the tail right of the m-th antecedent of any arbitrary sized implication chain. We further assume n > 1, since the wl-impchain-com-1.x 33870 series already covers the special case n = 1.

Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily.

The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 33870 series. (Contributed by Wolf Lammen, 17-Nov-2019.)

Theoremwl-impchain-com-2.3 33876 This theorem is in fact a copy of com23 86. It starts a series of theorems named after wl-impchain-com-n.m 33875. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜃 → (𝜓 → (𝜒𝜑)))

Theoremwl-impchain-com-2.4 33877 This theorem is in fact a copy of com24 95. It is another instantiation of theorems named after wl-impchain-com-n.m 33875. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜂 → (𝜃 → (𝜒 → (𝜓𝜑))))       (𝜂 → (𝜓 → (𝜒 → (𝜃𝜑))))

Theoremwl-impchain-com-3.2.1 33878 This theorem is in fact a copy of com3r 87. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 → (𝜒 → (𝜓𝜑)))       (𝜓 → (𝜃 → (𝜒𝜑)))

Theoremwl-impchain-a1-x 33879 If an implication chain is assumed (hypothesis) or proven (theorem) to hold, then we may add any extra antecedent to it, without changing its truth. This is expressed in its simplest form in wl-luk-a1i 33848, that allows us prepending an arbitrary antecedent to an implication chain. Using our antecedent swapping theorems described in wl-impchain-com-n.m 33875, we may then move such a prepended antecedent to any desired location within all antecedents. The first series of theorems of this kind adds a single antecedent somewhere to an implication chain. The appended number in the theorem name indicates its position within all antecedents, 1 denoting the head position. A second theorem series extends this idea to multiple additions (TODO).

Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one.

The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 50 in the main part.

The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 33870 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.)

Theoremwl-impchain-a1-1 33880 Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       (𝜓𝜑)

Theoremwl-impchain-a1-2 33881 Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))

Theoremwl-impchain-a1-3 33882 Inference rule, a copy of a1dd 50. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜃𝜒)))

20.17.3  An alternative axiom ~ ax-13

Axiomax-wl-13v 33883* A version of ax13v 2335 with a distinctor instead of a distinct variable expression.

Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1953. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremwl-ax13lem1 33884* A version of ax-wl-13v 33883 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

20.17.4  Other stuff

Theoremwl-mps 33885 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls1 33886 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)

Theoremwl-syls2 33887 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)       ((𝜓𝜒) → 𝜃)

Theoremwl-embant 33888 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-orel12 33889 In a conjunctive normal form a pair of nodes like (𝜑𝜓) ∧ (¬ 𝜑𝜒) eliminates the need of a node (𝜓𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)
(((𝜑𝜓) ∧ (¬ 𝜑𝜒)) → (𝜓𝜒))

Theoremwl-cases2-dnf 33890 A particular instance of orddi 995 and anddi 996 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1031, and is related to consensus 1036. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1047 and dfifp4 1050, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.)
(((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremwl-cbvmotv 33891* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑥⊤ → ∃*𝑦⊤)

Theoremwl-moteq 33892 Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑥⊤ → 𝑦 = 𝑧)

Theoremwl-motae 33893 Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)

Theoremwl-moae 33894* Two ways to express "at most one thing exists" or, in this context equivalently, "exactly one thing exists" . The equivalence results from the presence of ax-6 2021 in the proof, that ensures "at least one thing exists". For other equivalences see wl-euae 33895 and exists1 2692. Gerard Lang pointed out, that 𝑦𝑥𝑥 = 𝑦 with disjoint 𝑥 and 𝑦 (df-mo 2551, trut 1608) also means "exactly one thing exists" . (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies, and use ∃*. (Revised by Wolf Lammen, 7-Mar-2023.)
(∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremwl-euae 33895* Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremwl-nax6im 33896* The following series of theorems are centered around the empty domain, where no set exists. As a consequence, a set variable like 𝑥 has no instance to assign to. An expression like 𝑥 = 𝑦 is not really meaningful then. What does it evaluate to, true or false? In fact, the grammar extension weq 2005 requires us to formally assign a boolean value to an equation, say always false, unless you want to give up on exmid 881, for example. Whatever it is, we start out with the contraposition of ax-6 2021, that guarantees the existence of at least one set. Our hypothesis here expresses tentatively it might not hold. We can simplify the antecedent then, to the point where we do not need equation any more. This suggests what a decent characterization of the empty domain could be. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥 𝑥 = 𝑦𝜑)       (¬ ∃𝑥⊤ → 𝜑)

Theoremwl-nax6al 33897 In an empty domain the for-all operator always holds, even when applied to a false expression. This theorem actually shows that ax-5 1953 is provable there. Also we cannot assume that sp 2167 generally holds, except of course in the form of sptruw 1850. Without the support of an sp 2167 like theorem it seems difficult, if not impossible, to arrive at a theorem allowing to change the bounded variable in the antecedent. Additional axioms need to be postulated to further strengthen this result.

A consequence of this result is that 𝑥𝜑 is not true for any 𝜑. In particular, ∃*𝑥𝜑 does not hold either, a somewhat counterintuitive result. (Contributed by Wolf Lammen, 12-Mar-2023.)

(¬ ∃𝑥⊤ → ∀𝑥𝜑)

Theoremwl-nax6nfr 33898 All expressions are free of the variable used in the antecedent. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

Theoremwl-naev 33899* If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣)

Theoremwl-hbae1 33900 This specialization of hbae 2397 does not depend on ax-11 2150. (Contributed by Wolf Lammen, 8-Aug-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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