P. 374 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)

Type	Label	Description
Statement
21.20  Mathbox for Jim Kingdon
21.20.0.1  Circle constant
Theorem	taupilem3 37301	Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))
Theorem	taupilemrplb 37302*	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 37303	Lemma for taupi 37305. 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 37304	Lemma for taupi 37305. 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 37305	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)
21.20.0.2  Number theory
Theorem	dfgcd3 37306*	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))))
21.20.0.3  Real numbers
Theorem	irrdifflemf 37307	Lemma for irrdiff 37308. The forward direction. (Contributed by Jim Kingdon, 20-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ∈ ℚ)    &   ⊢ (𝜑 → 𝑅 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ≠ (abs‘(𝐴 − 𝑅)))
Theorem	irrdiff 37308*	The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.)
⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))))
Theorem	iccioo01 37309	The closed unit interval is equinumerous to the open unit interval. Based on a Mastodon post by Michael Kinyon. (Contributed by Jim Kingdon, 4-Jun-2024.)
⊢ (0[,]1) ≈ (0(,)1)
21.21  Mathbox for ML
21.21.1  Miscellaneous
Theorem	csbrecsg 37310	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 37311	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 37312*	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 37313*	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 37314	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (¬ 𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜓)
Theorem	con2bii2 37315	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ 𝜓)
Theorem	vtoclefex 37316*	Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜑)
Theorem	rnmptsn 37317*	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 37318*	This is the core of the proof of f1omptsn 37319, 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 37319*	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 37320*	This is the core of the proof of mptsnun 37321, 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 37321*	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 37322*	This is the core of the proof of dissneq 37323, 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 37323*	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 37324*	Closed form of exlimimd 37325. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)
Theorem	exlimimd 37325*	Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	exellim 37326*	Closed form of exellimddv 37327. See also exlimim 37324 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)
Theorem	exellimddv 37327*	Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 37326 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 37328*	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 37329*	This is the core of the proof of topdifinffin 37330, 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 37330*	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 37331*	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 37332*	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 37333*	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 37334	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 37335*	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 37336*	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 37337*	Lemma for isbasisrelowl 37340. (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 37338*	Lemma for isbasisrelowl 37340. (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 37339*	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 37340	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 37341	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 37342	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 37343*	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 37344*	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 37345	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 37346	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 37347	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 37348	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 37349	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 37350	The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.)
⊢ 1o = ∪ 2o
Theorem	rdgsucuni 37351	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 37352	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 37353	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8047. (Contributed by ML, 19-Oct-2020.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
Theorem	cbveud 37354*	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 37355*	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 37356	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 37357	Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.)
⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
Theorem	rdgellim 37358	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 37359	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 37360*	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 37361*	Lemma for exrecfn 37362. (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 37362*	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 37363*	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 37364	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))
21.21.2  Cartesian exponentiation
Syntax	cfinxp 37365	Extend the definition of a class to include Cartesian exponentiation.
class (𝑈↑↑𝑁)
Definition	df-finxp 37366*	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 8936 or df-map 8866 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 5148 can be used on it, and df-fv 6570 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 37374 and finxpsuc 37380 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 37367*	This theorem is the same as Definition df-finxp 37366, 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 37368	Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))
Theorem	finxpeq2 37369	Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))
Theorem	csbfinxpg 37370*	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 37371*	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 37372*	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 37373	The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)
⊢ (𝑈↑↑∅) = ∅
Theorem	finxp1o 37374	The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)
⊢ (𝑈↑↑1o) = 𝑈
Theorem	finxpreclem3 37375*	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 37376*	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 37377*	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 37378*	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 37379*	Lemma for finxpsuc 37380. (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 37380	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 37381	The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.)
⊢ (𝑈↑↑2o) = (𝑈 × 𝑈)
Theorem	finxp3o 37382	The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.)
⊢ (𝑈↑↑3o) = ((𝑈 × 𝑈) × 𝑈)
Theorem	finxpnom 37383	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 37384	Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)
⊢ (∅↑↑𝑁) = ∅
21.21.3  Topology
Theorem	iunctb2 37385	Using the axiom of countable choice ax-cc 10472, the countable union of countable sets is countable. See iunctb 10611 for a somewhat more general theorem. (Contributed by ML, 10-Dec-2020.)
⊢ (∀𝑥 ∈ ω 𝐵 ≼ ω → ∪ 𝑥 ∈ ω 𝐵 ≼ ω)
Theorem	domalom 37386*	A class which dominates every natural number is not finite. (Contributed by ML, 14-Dec-2020.)
⊢ (∀𝑛 ∈ ω 𝑛 ≼ 𝐴 → ¬ 𝐴 ∈ Fin)
Theorem	isinf2 37387*	The converse of isinf 9293. 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 37388*	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 37389*	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 37390*	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 37391*	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 37392*	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 37391 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 37393*	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 37394*	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)
21.21.3.1  Pi-base theorems 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 37396 defines countably compact topologies. Proofs of theorems are similarly labeled pibtN, for example pibt2 37399.
Theorem	pibp16 37395*	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 relabeled copy of iscmp 23411. (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 37396*	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 37397*	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 37398*	Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 37395 and pibp19 37396 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 37399*	Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 37396 and pibp21 37397 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)
21.22  Mathbox for Wolf Lammen
Theorem	wl-section-prop 37400	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 37401, ax-luk2 37402 and ax-luk3 37403. I rather copy this system than use luk-1 1651 to luk-3 1653, 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