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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-endmnd 34601 | The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.) |
⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd) | ||
Theorem | taupilem3 34602 | Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.) |
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1)) | ||
Theorem | taupilemrplb 34603* | A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.) |
⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦 | ||
Theorem | taupilem1 34604 | Lemma for taupi 34606. A positive real whose cosine is one is at least 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) |
⊢ ((𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1) → (2 · π) ≤ 𝐴) | ||
Theorem | taupilem2 34605 | Lemma for taupi 34606. The smallest positive real whose cosine is one is at most 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
⊢ τ ≤ (2 · π) | ||
Theorem | taupi 34606 | Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
⊢ τ = (2 · π) | ||
Theorem | dfgcd3 34607* | Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ0 ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)))) | ||
Theorem | csbdif 34608 | Distribution of class substitution over difference of two classes. (Contributed by ML, 14-Jul-2020.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∖ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∖ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | csbpredg 34609 | Move class substitution in and out of the predecessor class of a relationship. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋)) | ||
Theorem | csbwrecsg 34610 | Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹)) | ||
Theorem | csbrecsg 34611 | Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹)) | ||
Theorem | csbrdgg 34612 | Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼)) | ||
Theorem | csboprabg 34613* | Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈〈𝑦, 𝑧〉, 𝑑〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑑〉 ∣ [𝐴 / 𝑥]𝜑}) | ||
Theorem | csbmpo123 34614* | Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷)) | ||
Theorem | con1bii2 34615 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
⊢ (¬ 𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ ¬ 𝜓) | ||
Theorem | con2bii2 34616 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (¬ 𝜑 ↔ 𝜓) | ||
Theorem | vtoclefex 34617* | Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜑) | ||
Theorem | rnmptsn 34618* | The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.) |
⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | ||
Theorem | f1omptsnlem 34619* | This is the core of the proof of f1omptsn 34620, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) & ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} ⇒ ⊢ 𝐹:𝐴–1-1-onto→𝑅 | ||
Theorem | f1omptsn 34620* | 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 34621* | This is the core of the proof of mptsnun 34622, 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 34622* | 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 34623* | This is the core of the proof of dissneq 34624, 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 34624* | 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 34625* | Closed form of exlimimd 34626. (Contributed by ML, 17-Jul-2020.) |
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓) | ||
Theorem | exlimimd 34626* | Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | exellim 34627* | Closed form of exellimddv 34628. See also exlimim 34625 for a more general theorem. (Contributed by ML, 17-Jul-2020.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) | ||
Theorem | exellimddv 34628* | Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 34627 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 34629* | 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 34630* | This is the core of the proof of topdifinffin 34631, 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 34631* | 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 34632* | 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 34633* | 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 34634* | 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 34635 | 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 34636* | 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 34637* | 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 34638* | Lemma for isbasisrelowl 34641. (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 34639* | Lemma for isbasisrelowl 34641. (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 34640* | 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 34641 | 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 34642 | 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 34643 | 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 34644* | 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 34645* | 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 34646 | 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 34647 | 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 34648 | 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 34649 | 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 34650 | 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 34651 | The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.) |
⊢ 1o = ∪ 2o | ||
Theorem | rdgsucuni 34652 | 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 34653 | 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 34654 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7726. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) | ||
Theorem | cbveud 34655* | 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 34656* | 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 34657 | 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 34658 | Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.) |
⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) | ||
Theorem | rdgellim 34659 | 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 34660 | 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 34661* | 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 34662* | Lemma for exrecfn 34663. (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 34663* | 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 34664* | 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 34665 | 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 34666 | Extend the definition of a class to include Cartesian exponentiation. |
class (𝑈↑↑𝑁) | ||
Definition | df-finxp 34667* |
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 8464 or df-map 8410 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 5069 can be used on it, and df-fv 6365 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 34675 and finxpsuc 34681 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 34668* | This theorem is the same as the definition df-finxp 34667, 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 34669 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁)) | ||
Theorem | finxpeq2 34670 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁)) | ||
Theorem | csbfinxpg 34671* | 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 34672* | 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 34673* | 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 34674 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) |
⊢ (𝑈↑↑∅) = ∅ | ||
Theorem | finxp1o 34675 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) |
⊢ (𝑈↑↑1o) = 𝑈 | ||
Theorem | finxpreclem3 34676* | 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 34677* | 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 34678* | 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 34679* | 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 34680* | Lemma for finxpsuc 34681. (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 34681 | 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 34682 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | ||
Theorem | finxp3o 34683 | The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) |
⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) | ||
Theorem | finxpnom 34684 | 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 34685 | Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) |
⊢ (∅↑↑𝑁) = ∅ | ||
Theorem | iunctb2 34686 | Using the axiom of countable choice ax-cc 9859, the countable union of countable sets is countable. See iunctb 9998 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.) |
⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) | ||
Theorem | domalom 34687* | A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.) |
⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin) | ||
Theorem | isinf2 34688* | The converse of isinf 8733. 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 34689* | 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 34690* | 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 34691* | 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 34692* | 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 34693* | 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 34692 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 34694* | 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 34695* | 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 34697 defines countably compact topologies. Proofs of theorems are similarly labelled pibtN, for example pibt2 34700. | ||
Theorem | pibp16 34696* | 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 21998. (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 34697* | 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 34698* | 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 34699* | Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 34696 and pibp19 34697 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 34700* | Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 34697 and pibp21 34698 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) |
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