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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | taupilem1 34601 | Lemma for taupi 34603. 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 34602 | Lemma for taupi 34603. 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 34603 | 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 34604* | 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 34605 | 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 34606 | 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 34607 | 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 34608 | 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 34609 | 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 34610* | 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 34611* | 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 34612 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
⊢ (¬ 𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ ¬ 𝜓) | ||
Theorem | con2bii2 34613 | A contraposition inference. (Contributed by ML, 18-Oct-2020.) |
⊢ (𝜑 ↔ ¬ 𝜓) ⇒ ⊢ (¬ 𝜑 ↔ 𝜓) | ||
Theorem | vtoclefex 34614* | Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝐴 → 𝜑) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜑) | ||
Theorem | rnmptsn 34615* | 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 34616* | This is the core of the proof of f1omptsn 34617, 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 34617* | 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 34618* | This is the core of the proof of mptsnun 34619, 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 34619* | 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 34620* | This is the core of the proof of dissneq 34621, 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 34621* | 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 34622* | Closed form of exlimimd 34623. (Contributed by ML, 17-Jul-2020.) |
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓) | ||
Theorem | exlimimd 34623* | Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | exellim 34624* | Closed form of exellimddv 34625. See also exlimim 34622 for a more general theorem. (Contributed by ML, 17-Jul-2020.) |
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑) | ||
Theorem | exellimddv 34625* | Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 34624 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 34626* | 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 34627* | This is the core of the proof of topdifinffin 34628, 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 34628* | 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 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 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 34630* | 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 34631* | 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 34632 | 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 34633* | 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 34634* | 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 34635* | Lemma for isbasisrelowl 34638. (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 34636* | Lemma for isbasisrelowl 34638. (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 34637* | 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 34638 | 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 34639 | 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 34640 | 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 34641* | 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 34642* | 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 34643 | 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 34644 | 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 34645 | 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 34646 | 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 34647 | 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 34648 | The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.) |
⊢ 1o = ∪ 2o | ||
Theorem | rdgsucuni 34649 | 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 34650 | 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 34651 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7723. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) | ||
Theorem | cbveud 34652* | 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 34653* | 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 34654 | 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 34655 | Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.) |
⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶))) | ||
Theorem | rdgellim 34656 | 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 34657 | 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 34658* | 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 34659* | Lemma for exrecfn 34660. (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 34660* | 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 34661* | 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 34662 | 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 34663 | Extend the definition of a class to include Cartesian exponentiation. |
class (𝑈↑↑𝑁) | ||
Definition | df-finxp 34664* |
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 8461 or df-map 8407 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 5066 can be used on it, and df-fv 6362 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 34672 and finxpsuc 34678 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 34665* | This theorem is the same as the definition df-finxp 34664, 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 34666 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁)) | ||
Theorem | finxpeq2 34667 | Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁)) | ||
Theorem | csbfinxpg 34668* | 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 34669* | 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 34670* | 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 34671 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) |
⊢ (𝑈↑↑∅) = ∅ | ||
Theorem | finxp1o 34672 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) |
⊢ (𝑈↑↑1o) = 𝑈 | ||
Theorem | finxpreclem3 34673* | 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 34674* | 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 34675* | 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 34676* | 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 34677* | Lemma for finxpsuc 34678. (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 34678 | 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 34679 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) |
⊢ (𝑈↑↑2o) = (𝑈 × 𝑈) | ||
Theorem | finxp3o 34680 | The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) |
⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈) | ||
Theorem | finxpnom 34681 | 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 34682 | Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) |
⊢ (∅↑↑𝑁) = ∅ | ||
Theorem | iunctb2 34683 | Using the axiom of countable choice ax-cc 9856, the countable union of countable sets is countable. See iunctb 9995 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.) |
⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω) | ||
Theorem | domalom 34684* | A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.) |
⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin) | ||
Theorem | isinf2 34685* | The converse of isinf 8730. 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 34686* | 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 34687* | 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 34688* | 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 34689* | 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 34690* | 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 34689 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 34691* | 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 34692* | 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 34694 defines countably compact topologies. Proofs of theorems are similarly labelled pibtN, for example pibt2 34697. | ||
Theorem | pibp16 34693* | 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 21995. (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 34694* | 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 34695* | 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 34696* | Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 34693 and pibp19 34694 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 34697* | Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 34694 and pibp21 34695 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 34698 |
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 34699, ax-luk2 34700 and ax-luk3 34701. I rather copy this system than use luk-1 1652 to luk-3 1654, 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 34699 |
1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-1 1652 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 34700 |
2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-2 1653 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 pm2.65 195):
((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) |
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