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Type | Label | Description | ||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||
Theorem | f1omptsn 34501* | A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ 𝐹:𝐴–1-1-onto→𝑅 | ||||||||||||||||||||||||
Theorem | mptsnunlem 34502* | This is the core of the proof of mptsnun 34503, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) | ||||||||||||||||||||||||
Theorem | mptsnun 34503* | 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 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵)) | ||||||||||||||||||||||||
Theorem | dissneqlem 34504* | This is the core of the proof of dissneq 34505, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴) | ||||||||||||||||||||||||
Theorem | dissneq 34505* | Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴) | ||||||||||||||||||||||||
Theorem | exlimim 34506* | Closed form of exlimimd 34507. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓) | ||||||||||||||||||||||||
Theorem | exlimimd 34507* | Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | exellim 34508* | Closed form of exellimddv 34509. See also exlimim 34506 for a more general theorem. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) | ||||||||||||||||||||||||
Theorem | exellimddv 34509* | Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 34508 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.) | ||||||||||||||||||||||
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | topdifinfindis 34510* | 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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴}) | ||||||||||||||||||||||||
Theorem | topdifinffinlem 34511* | This is the core of the proof of topdifinffin 34512, 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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | topdifinffin 34512* | 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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | topdifinf 34513* | 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 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} ⇒ ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴})) | ||||||||||||||||||||||||
Theorem | topdifinfeq 34514* | Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.) | ||||||||||||||||||||||
⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))} | ||||||||||||||||||||||||
Theorem | icorempo 34515* | Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = ([,) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | ||||||||||||||||||||||||
Theorem | icoreresf 34516 | Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.) | ||||||||||||||||||||||
⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ | ||||||||||||||||||||||||
Theorem | icoreval 34517* | 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 ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)}) | ||||||||||||||||||||||||
Theorem | icoreelrnab 34518* | Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | ||||||||||||||||||||||||
Theorem | isbasisrelowllem1 34519* | Lemma for isbasisrelowl 34522. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | isbasisrelowllem2 34520* | Lemma for isbasisrelowl 34522. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | icoreclin 34521* | The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.) | ||||||||||||||||||||||
⊢ 𝐼 = ([,) “ (ℝ × ℝ)) ⇒ ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) | ||||||||||||||||||||||||
Theorem | isbasisrelowl 34522 | 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 34523 | 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 34524 | 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 34525* | 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 34526* | 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 34527 | 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 34528 | 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 34529 | 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 34530 | 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 34531 | 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 34532 | The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ 1o = ∪ 2o | ||||||||||||||||||||||||
Theorem | rdgsucuni 34533 | 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 34534 | 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 34535 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7715. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) | ||||||||||||||||||||||||
Theorem | cbveud 34536* | Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒)) | ||||||||||||||||||||||||
Theorem | cbvreud 34537* | Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒)) | ||||||||||||||||||||||||
Theorem | difunieq 34538 | The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.) | ||||||||||||||||||||||
⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵) | ||||||||||||||||||||||||
Theorem | inunissunidif 34539 | Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.) | ||||||||||||||||||||||
⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) | ||||||||||||||||||||||||
Theorem | rdgellim 34540 | Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.) | ||||||||||||||||||||||
⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵))) | ||||||||||||||||||||||||
Theorem | rdglimss 34541 | A recursive definition at a limit ordinal is a superset of itself at any smaller ordinal. (Contributed by ML, 30-Mar-2022.) | ||||||||||||||||||||||
⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵)) | ||||||||||||||||||||||||
Theorem | rdgssun 34542* | In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵)) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋)) | ||||||||||||||||||||||||
Theorem | exrecfnlem 34543* | Lemma for exrecfn 34544. (Contributed by ML, 30-Mar-2022.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵))) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥)) | ||||||||||||||||||||||||
Theorem | exrecfn 34544* | Theorem about the existence of infinite recursive sets. 𝑦 should usually be free in 𝐵. (Contributed by ML, 30-Mar-2022.) | ||||||||||||||||||||||
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥)) | ||||||||||||||||||||||||
Theorem | exrecfnpw 34545* | For any base set, a set which contains the powerset of all of its own elements exists. (Contributed by ML, 30-Mar-2022.) | ||||||||||||||||||||||
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥)) | ||||||||||||||||||||||||
Theorem | finorwe 34546 | If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.) | ||||||||||||||||||||||
⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) | ||||||||||||||||||||||||
Syntax | cfinxp 34547 | Extend the definition of a class to include Cartesian exponentiation. | ||||||||||||||||||||||
class (𝑈↑↑𝑁) | ||||||||||||||||||||||||
Definition | df-finxp 34548* |
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 8451 or df-map 8398 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 5059 can be used on it, and df-fv 6357 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 34556 and finxpsuc 34562 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))} | ||||||||||||||||||||||||
Theorem | dffinxpf 34549* | This theorem is the same as the definition df-finxp 34548, 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 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} | ||||||||||||||||||||||||
Theorem | finxpeq1 34550 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁)) | ||||||||||||||||||||||||
Theorem | finxpeq2 34551 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁)) | ||||||||||||||||||||||||
Theorem | csbfinxpg 34552* | 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 34553* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) | ||||||||||||||||||||||||
Theorem | finxpreclem2 34554* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) | ||||||||||||||||||||||||
Theorem | finxp0 34555 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑∅) = ∅ | ||||||||||||||||||||||||
Theorem | finxp1o 34556 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑1o) = 𝑈 | ||||||||||||||||||||||||
Theorem | finxpreclem3 34557* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 〈∪ 𝑁, (1st ‘𝑋)〉 = (𝐹‘〈𝑁, 𝑋〉)) | ||||||||||||||||||||||||
Theorem | finxpreclem4 34558* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁) = (rec(𝐹, 〈∪ 𝑁, (1st ‘𝑦)〉)‘∪ 𝑁)) | ||||||||||||||||||||||||
Theorem | finxpreclem5 34559* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑛 ∈ ω ∧ 1o ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘〈𝑛, 𝑥〉) = 〈𝑛, 𝑥〉)) | ||||||||||||||||||||||||
Theorem | finxpreclem6 34560* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1o ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)) | ||||||||||||||||||||||||
Theorem | finxpsuclem 34561* | Lemma for finxpsuc 34562. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) | ||||||||||||||||||||||||
Theorem | finxpsuc 34562 | 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 34563 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | ||||||||||||||||||||||||
Theorem | finxp3o 34564 | The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) | ||||||||||||||||||||||||
Theorem | finxpnom 34565 | 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 34566 | Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (∅↑↑𝑁) = ∅ | ||||||||||||||||||||||||
Theorem | iunctb2 34567 | Using the axiom of countable choice ax-cc 9846, the countable union of countable sets is countable. See iunctb 9985 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.) | ||||||||||||||||||||||
⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) | ||||||||||||||||||||||||
Theorem | domalom 34568* | A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.) | ||||||||||||||||||||||
⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | isinf2 34569* | The converse of isinf 8720. Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by ML, 14-Dec-2020.) | ||||||||||||||||||||||
⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ¬ 𝐴 ∈ Fin) | ||||||||||||||||||||||||
Theorem | ctbssinf 34570* | Using the axiom of choice, any infinite class has a countable subset. (Contributed by ML, 14-Dec-2020.) | ||||||||||||||||||||||
⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω)) | ||||||||||||||||||||||||
Theorem | ralssiun 34571* | The index set of an indexed union is a subset of the union when each 𝐵 contains its index. (Contributed by ML, 16-Dec-2020.) | ||||||||||||||||||||||
⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||||||||||||||||||||||||
Theorem | nlpineqsn 34572* | For every point 𝑝 of a subset 𝐴 of 𝑋 with no limit points, there exists an open set 𝑛 that intersects 𝐴 only at 𝑝. (Contributed by ML, 23-Mar-2021.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∀𝑝 ∈ 𝐴 ∃𝑛 ∈ 𝐽 (𝑝 ∈ 𝑛 ∧ (𝑛 ∩ 𝐴) = {𝑝})) | ||||||||||||||||||||||||
Theorem | nlpfvineqsn 34573* | Given a subset 𝐴 of 𝑋 with no limit points, there exists a function from each point 𝑝 of 𝐴 to an open set intersecting 𝐴 only at 𝑝. This proof uses the axiom of choice. (Contributed by ML, 23-Mar-2021.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝}))) | ||||||||||||||||||||||||
Theorem | fvineqsnf1 34574* | A theorem about functions where the image of every point intersects the domain only at that point. If 𝐽 is a topology and 𝐴 is a set with no limit points, then there exists an 𝐹 such that this antecedent is true. See nlpfvineqsn 34573 for a proof of this fact. (Contributed by ML, 23-Mar-2021.) | ||||||||||||||||||||||
⊢ ((𝐹:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→𝐽) | ||||||||||||||||||||||||
Theorem | fvineqsneu 34575* | A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 27-Mar-2021.) | ||||||||||||||||||||||
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥) | ||||||||||||||||||||||||
Theorem | fvineqsneq 34576* | A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 28-Mar-2021.) | ||||||||||||||||||||||
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹 ∧ 𝐴 ⊆ ∪ 𝑍)) → 𝑍 = ran 𝐹) | ||||||||||||||||||||||||
This section contains a few proofs of theorems found in the pi-base database. The pi-base site can be found at <https://topology.pi-base.org/>. Definitions of topological properties are theorems labeled pibpN, where N is the property number in pi-base. For example, pibp19 34578 defines countably compact topologies. Proofs of theorems are similarly labelled pibtN, for example pibt2 34581. | ||||||||||||||||||||||||
Theorem | pibp16 34577* | Property P000016 of pi-base. The class of compact topologies. A space 𝑋 is compact if every open cover of 𝑋 has a finite subcover. This theorem is just a relabelled copy of iscmp 21926. (Contributed by ML, 8-Dec-2020.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) | ||||||||||||||||||||||||
Theorem | pibp19 34578* | Property P000019 of pi-base. The class of countably compact topologies. A space 𝑋 is countably compact if every countable open cover of 𝑋 has a finite subcover. (Contributed by ML, 8-Dec-2020.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} ⇒ ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) | ||||||||||||||||||||||||
Theorem | pibp21 34579* | Property P000021 of pi-base. The class of weakly countably compact topologies, or limit point compact topologies. A space 𝑋 is weakly countably compact if every infinite subset of 𝑋 has a limit point. (Contributed by ML, 9-Dec-2020.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)} ⇒ ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦))) | ||||||||||||||||||||||||
Theorem | pibt1 34580* | Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 34577 and pibp19 34578 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.) | ||||||||||||||||||||||
⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} ⇒ ⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶) | ||||||||||||||||||||||||
Theorem | pibt2 34581* | Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 34578 and pibp21 34579 for the definitions of the relevant properties. This proof uses the axiom of choice. (Contributed by ML, 30-Mar-2021.) | ||||||||||||||||||||||
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)} & ⊢ 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)} ⇒ ⊢ (𝐽 ∈ 𝐶 → 𝐽 ∈ 𝑊) | ||||||||||||||||||||||||
Theorem | wl-section-prop 34582 |
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 34583, ax-luk2 34584 and ax-luk3 34585. I rather copy this system than use luk-1 1647 to luk-3 1649, 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 34583 |
1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-1 1647 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 34584 |
2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-2 1648 or pm2.18 128, 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 pm2.65 194):
((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||||||||||||||||||||||||
Axiom | ax-luk3 34585 |
3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-3 1649 and pm2.24 124, but introduced as an axiom.
One might think that the similar pm2.21 123 (¬ 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 34583, ax-luk2 34584
and pm2.21 123 result in phi1 or phi2, meaning they always hold. But for
wl-luk-ax3 34597 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 ----------------------------------------------------------------------
Implication is defined as ----------------------------------------------------------------------
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-section-boot 34586 | 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 34587 | 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 34588 | An inference version of the transitive laws for implication luk-1 1647. 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-imtrid 34589 | 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 34590 | 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 34591 | 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 34592 | Inference rule. Copy of pm2.24i 153 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 34593 | 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 34594 | 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 34595 | 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-imtrdi 34596 | 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 34597 | 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 34598 | 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 34599 | 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 34600 | 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)) |
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