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Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.16  Mathbox for ML

Theoremcsbdif 33301 Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Theoremcsbpredg 33302 Move class substitution in and out of the predecessor class of a relationship. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥Pred(𝑅, 𝐷, 𝑋) = Pred(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝑋))

Theoremcsbwrecsg 33303 Move class substitution in and out of the well-founded recursive function generator . (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥wrecs(𝑅, 𝐷, 𝐹) = wrecs(𝐴 / 𝑥𝑅, 𝐴 / 𝑥𝐷, 𝐴 / 𝑥𝐹))

Theoremcsbrecsg 33304 Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥recs(𝐹) = recs(𝐴 / 𝑥𝐹))

Theoremcsbrdgg 33305 Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥rec(𝐹, 𝐼) = rec(𝐴 / 𝑥𝐹, 𝐴 / 𝑥𝐼))

Theoremcsboprabg 33306* Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})

Theoremcsbmpt22g 33307* Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌, 𝑧𝑍𝐷) = (𝑦𝐴 / 𝑥𝑌, 𝑧𝐴 / 𝑥𝑍𝐴 / 𝑥𝐷))

Theoremmpnanrd 33308 Eliminate the right side of a negated conjunction in an implication. (Contributed by ML, 17-Oct-2020.)
(𝜑𝜓)    &   (𝜑 → ¬ (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremcon1bii2 33309 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
𝜑𝜓)       (𝜑 ↔ ¬ 𝜓)

Theoremcon2bii2 33310 A contraposition inference. (Contributed by ML, 18-Oct-2020.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)

Theoremvtoclefex 33311* Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
𝑥𝜑    &   (𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)

Theoremrnmptsn 33312* The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
ran (𝑥𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}

Theoremf1omptsnlem 33313* This is the core of the proof of f1omptsn 33314, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅

Theoremf1omptsn 33314* A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       𝐹:𝐴1-1-onto𝑅

Theoremmptsnunlem 33315* This is the core of the proof of mptsnun 33316, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))

Theoremmptsnun 33316* A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.)
𝐹 = (𝑥𝐴 ↦ {𝑥})    &   𝑅 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       (𝐵𝐴𝐵 = (𝐹𝐵))

Theoremdissneqlem 33317* This is the core of the proof of dissneq 33318, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Theoremdissneq 33318* Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
𝐶 = {𝑢 ∣ ∃𝑥𝐴 𝑢 = {𝑥}}       ((𝐶𝐵𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Theoremexlimim 33319* Closed form of exlimimd 33320. (Contributed by ML, 17-Jul-2020.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)

Theoremexlimimd 33320* Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremexlimimdd 33321 Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremexellim 33322* Closed form of exellimddv 33323. See also exlimim 33319 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)

Theoremexellimddv 33323* Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 33322 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)
(𝜑 → ∃𝑥 𝑥𝐴)    &   (𝜑 → (𝑥𝐴𝜓))       (𝜑𝜓)

Theoremtopdifinfindis 33324* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})

Theoremtopdifinffinlem 33325* This is the core of the proof of topdifinffin 33326, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Theoremtopdifinffin 33326* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)

Theoremtopdifinf 33327* Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.)
𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}       ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))

Theoremtopdifinfeq 33328* Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.)
{𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ ((𝐴𝑥) = ∅ ∨ (𝐴𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}

Theoremicorempt2 33329* Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
𝐹 = ([,) ↾ (ℝ × ℝ))       𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥𝑧𝑧 < 𝑦)})

Theoremicoreresf 33330 Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.)
([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ

Theoremicoreval 33331* Value of the closed-below, open-above interval function on reals. (Contributed by ML, 26-Jul-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴𝑧𝑧 < 𝐵)})

Theoremicoreelrnab 33332* Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       (𝑋𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)})

Theoremisbasisrelowllem1 33333* Lemma for isbasisrelowl 33336. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑏𝑑)) → (𝑥𝑦) ∈ 𝐼)

Theoremisbasisrelowllem2 33334* Lemma for isbasisrelowl 33336. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎𝑧𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐𝑧𝑧 < 𝑑)})) ∧ (𝑎𝑐𝑑𝑏)) → (𝑥𝑦) ∈ 𝐼)

Theoremicoreclin 33335* The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)
𝐼 = ([,) “ (ℝ × ℝ))       ((𝑥𝐼𝑦𝐼) → (𝑥𝑦) ∈ 𝐼)

Theoremisbasisrelowl 33336 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 33337 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 33338 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 33339* 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 33340* 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 33341 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 33342 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 33343 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 33344 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 33345 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 33346 The ordinal number 1𝑜 is the predecessor of the ordinal number 2𝑜. (Contributed by ML, 19-Oct-2020.)
1𝑜 = 2𝑜

Theoremrdgsucuni 33347 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 33348 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 33349 Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7245. (Contributed by ML, 19-Oct-2020.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))

Syntaxcfinxp 33350 Extend the definition of a class to include Cartesian exponentiation.
class (𝑈↑↑𝑁)

Definitiondf-finxp 33351* 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 7951 or df-map 7901 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 4686 can be used on it, and df-fv 5934 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 33359 and finxpsuc 33365 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 33352* This theorem is the same as the definition df-finxp 33351, 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 33353 Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
(𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Theoremfinxpeq2 33354 Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
(𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Theoremcsbfinxpg 33355* Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥(𝑈↑↑𝑁) = (𝐴 / 𝑥𝑈↑↑𝐴 / 𝑥𝑁))

Theoremfinxpreclem1 33356* 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 33357* 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 33358 The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑∅) = ∅

Theoremfinxp1o 33359 The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)
(𝑈↑↑1𝑜) = 𝑈

Theoremfinxpreclem3 33360* 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 33361* 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 33362* 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 33363* 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 33364* Lemma for finxpsuc 33365. (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 33365 The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.)
((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Theoremfinxp2o 33366 The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)
(𝑈↑↑2𝑜) = (𝑈 × 𝑈)

Theoremfinxp3o 33367 The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)
(𝑈↑↑3𝑜) = ((𝑈 × 𝑈) × 𝑈)

Theoremfinxpnom 33368 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 33369 Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)
(∅↑↑𝑁) = ∅

20.17  Mathbox for Wolf Lammen

Theoremwl-section-prop 33370 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 33371, ax-luk2 33372 and ax-luk3 33373. I rather copy this system than use luk-1 1620 to luk-3 1622, 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 33371 1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-1 1620 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 33372 2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-2 1621 or pm2.18 122, 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 pm2.65 184):

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

((¬ 𝜑𝜑) → 𝜑)

Axiomax-luk3 33373 3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 1622 and pm2.24 121, but introduced as an axiom. One might think that the similar pm2.21 120 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 33371, ax-luk2 33372 and pm2.21 120 result in phi1 or phi2, meaning they always hold. But for wl-ax3 33385 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 33374 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-imim1i 33375 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-syl 33376 An inference version of the transitive laws for implication luk-1 1620. 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-syl5 33377 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-pm2.18d 33378 Deduction based on reductio ad absurdum. Copy of pm2.18d 124 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)

Theoremwl-con4i 33379 Inference rule. Copy of con4i 113 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theoremwl-pm2.24i 33380 Inference rule. Copy of pm2.24i 146 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       𝜑𝜓)

Theoremwl-a1i 33381 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-mpi 33382 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-imim2i 33383 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-syl6 33384 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-ax3 33385 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-ax1 33386 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-pm2.27 33387 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-com12 33388 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-pm2.21 33389 From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 120 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))

Theoremwl-con1i 33390 A contraposition inference. Copy of con1i 144 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜓)       𝜓𝜑)

Theoremwl-ja 33391 Inference joining the antecedents of two premises. Copy of ja 173 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑𝜒)    &   (𝜓𝜒)       ((𝜑𝜓) → 𝜒)

Theoremwl-imim2 33392 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-a1d 33393 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-ax2 33394 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-id 33395 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-notnotr 33396 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 125 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremwl-pm2.04 33397 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 33398 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 33399 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 33400 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

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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