P. 339 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

isbasisrelowl 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

Theorem

icoreunrn 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

Theorem

istoprelowl 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)

Theorem

icoreelrn 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 ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼)

Theorem

iooelexlt 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)

Theorem

relowlssretop 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)

Theorem

relowlpssretop 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)

Theorem

sucneqond 33808

Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)

⊢ (𝜑 → 𝑋 = suc 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ On)    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Theorem

sucneqoni 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    ⇒   ⊢ 𝑋 ≠ 𝑌

Theorem

onsucuni3 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)

Theorem

1oequni2o 33811

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

⊢ 1_o = ∪ 2_o

Theorem

rdgsucuni 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)))

Theorem

rdgeqoa 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))))

Theorem

elxp8 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) ↔ ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Syntax

cfinxp 33815

Extend the definition of a class to include Cartesian exponentiation.

class (𝑈↑↑𝑁)

Definition

df-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↑↑2_o), 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 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}

Theorem

dffinxpf 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 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}

Theorem

finxpeq1 33818

Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Theorem

finxpeq2 33819

Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Theorem

csbfinxpg 33820*

Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)

⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))

Theorem

finxpreclem1 33821*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_o, 𝑋⟩))

Theorem

finxpreclem2 33822*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_o, 𝑋⟩))

Theorem

finxp0 33823

The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)

⊢ (𝑈↑↑∅) = ∅

Theorem

finxp1o 33824

The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)

⊢ (𝑈↑↑1_o) = 𝑈

Theorem

finxpreclem3 33825*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2_o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Theorem

finxpreclem4 33826*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1^st ‘𝑦)⟩)‘∪ 𝑁))

Theorem

finxpreclem5 33827*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑛 ∈ ω ∧ 1_o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Theorem

finxpreclem6 33828*

Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 1_o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈))

Theorem

finxpsuclem 33829*

Lemma for finxpsuc 33830. (Contributed by ML, 24-Oct-2020.)

⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theorem

finxpsuc 33830

The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.)

⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theorem

finxp2o 33831

The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)

⊢ (𝑈↑↑2_o) = (𝑈 × 𝑈)

Theorem

finxp3o 33832

The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)

⊢ (𝑈↑↑3_o) = ((𝑈 × 𝑈) × 𝑈)

Theorem

finxpnom 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) = ∅)

Theorem

finxp00 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.

Theorem

cnfin0 33835

The empty set is an ordinal in Cantor normal form. (Contributed by ML, 24-Jun-2022.)

⊢ 𝐼 = {⟨∅, 1_o⟩}    &   ⊢ + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦‘𝑛) +_o (𝑧‘𝑛))))    &   ⊢ (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))    &   ⊢ (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))    &   ⊢ (𝜒 ↔ ∃𝑎∃𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1_o⟩} ∧ 𝑦 = {⟨𝑏, 1_o⟩})))    &   ⊢ 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓 ∨ 𝜒)))}    &   ⊢ < = ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)    &   ⊢ 𝐶 = dom <    ⇒   ⊢ ∅ ∈ 𝐶

Theorem

cnfinltrel 33836*

Less than for the Cantor normal form is a relation. (Contributed by ML, 24-Jun-2022.)

⊢ 𝐼 = {⟨∅, 1_o⟩}    &   ⊢ + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦‘𝑛) +_o (𝑧‘𝑛))))    &   ⊢ (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))    &   ⊢ (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))    &   ⊢ (𝜒 ↔ ∃𝑎∃𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1_o⟩} ∧ 𝑦 = {⟨𝑏, 1_o⟩})))    &   ⊢ 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓 ∨ 𝜒)))}    &   ⊢ < = ∪ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)    &   ⊢ 𝐶 = dom <    ⇒   ⊢ Rel <

20.17  Mathbox for Wolf Lammen

Theorem

wl-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

Axiom

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

⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))

Axiom

ax-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):

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

⊢ ((¬ 𝜑 → 𝜑) → 𝜑)

Axiom

ax-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 → (𝜑 → 𝜓)) 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 → (¬ 𝜑 → 𝜓))

20.17.1  1. Bootstrapping

Theorem

wl-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

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒))

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))

Theorem

wl-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 → (¬ 𝜓 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)

Theorem

wl-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)    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-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    ⇒   ⊢ (¬ 𝜑 → 𝜓)

Theorem

wl-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    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-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 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜓))

Theorem

wl-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 → (𝜓 → 𝜒))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))

Theorem

wl-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) → (𝜓 → 𝜑))

Theorem

wl-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 → (𝜓 → 𝜑))

Theorem

wl-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 → ((𝜑 → 𝜓) → 𝜓))

Theorem

wl-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 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 → 𝜒))

Theorem

wl-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 → (𝜑 → 𝜓))

Theorem

wl-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.)

⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ (¬ 𝜓 → 𝜑)

Theorem

wl-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.)

⊢ (¬ 𝜑 → 𝜒)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜒)

Theorem

wl-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.)

⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))

Theorem

wl-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 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))

Theorem

wl-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.)

⊢ (𝜑 → 𝜑)

Theorem

wl-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.)

⊢ (¬ ¬ 𝜑 → 𝜑)

Theorem

wl-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 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))

20.17.2  Implication chains

Theorem

wl-section-impchain 33865

An implication like (𝜓 → 𝜑) with one antecedent can easily be extended by prepending more and more antecedents, as in (𝜒 → (𝜓 → 𝜑)) or (𝜃 → (𝜒 → (𝜓 → 𝜑))). 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

Theorem

wl-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.)

⊢ ⊤

Theorem

wl-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    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ 𝜑

Theorem

wl-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.)

⊢ (𝜒 → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜒 → 𝜑)

Theorem

wl-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 → (𝜒 → 𝜓))    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜃 → (𝜒 → 𝜑))

Theorem

wl-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.)

⊢ ⊤

Theorem

wl-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.)

⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-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 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜑))

Theorem

wl-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 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜑)))

Theorem

wl-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 → (𝜓 → 𝜑))))    ⇒   ⊢ (𝜓 → (𝜃 → (𝜒 → (𝜂 → 𝜑))))

Theorem

wl-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.)

⊢ ⊤

Theorem

wl-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 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜃 → (𝜓 → (𝜒 → 𝜑)))

Theorem

wl-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 → (𝜓 → 𝜑))))    ⇒   ⊢ (𝜂 → (𝜓 → (𝜒 → (𝜃 → 𝜑))))

Theorem

wl-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 → (𝜓 → 𝜑)))    ⇒   ⊢ (𝜓 → (𝜃 → (𝜒 → 𝜑)))

Theorem

wl-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.)

⊢ ⊤

Theorem

wl-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    ⇒   ⊢ (𝜓 → 𝜑)

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))

Theorem

wl-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 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

20.17.3  An alternative axiom ~ ax-13

Axiom

ax-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))

Theorem

wl-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

Theorem

wl-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 → (𝜓 → 𝜒))    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜃)

Theorem

wl-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.)

⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜃)

Theorem

wl-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.)

⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 → 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → 𝜃)

Theorem

wl-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    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ ((𝜑 → 𝜓) → 𝜒)

Theorem

wl-orel12 33889

In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.)

⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒))

Theorem

wl-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.)

⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒)))

Theorem

wl-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⊤)

Theorem

wl-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)

Theorem

wl-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)

Theorem

wl-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 ∃𝑦∀𝑥𝑥 = 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)

Theorem

wl-euae 33895*

Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)

⊢ (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theorem

wl-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 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∃𝑥⊤ → 𝜑)

Theorem

wl-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)

Theorem

wl-nax6nfr 33898

All expressions are free of the variable used in the antecedent. (Contributed by Wolf Lammen, 12-Mar-2023.)

⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑)

Theorem

wl-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)

Theorem

wl-hbae1 33900

This specialization of hbae 2397 does not depend on ax-11 2150. (Contributed by Wolf Lammen, 8-Aug-2021.)

⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦)