P. 65 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elsuci 6401	Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 6-Jun-1994.)
⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsucg 6402	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc2g 6403	Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc 6404	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsuc2 6405	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
Theorem	nfsuc 6406	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 suc 𝐴
Theorem	elelsuc 6407	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)
Theorem	sucel 6408*	Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
⊢ (suc 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
Theorem	suc0 6409	The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
⊢ suc ∅ = {∅}
Theorem	sucprc 6410	A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
Theorem	unisucs 6411	The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6413. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
Theorem	unisucg 6412	A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6413. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
Theorem	unisuc 6413	A transitive class is equal to the union of its successor, inference form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)
Theorem	sssucid 6414	A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
⊢ 𝐴 ⊆ suc 𝐴
Theorem	sucidg 6415	Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). Lemma 1.7 of [Schloeder] p. 1. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
Theorem	sucid 6416	A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ suc 𝐴
Theorem	nsuceq0 6417	No successor is empty. (Contributed by NM, 3-Apr-1995.)
⊢ suc 𝐴 ≠ ∅
Theorem	eqelsuc 6418	A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)
Theorem	iunsuc 6419*	Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ suc 𝐴𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)
Theorem	suctr 6420	The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ (Tr 𝐴 → Tr suc 𝐴)
Theorem	trsuc 6421	A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
Theorem	trsucss 6422	A member of the successor of a transitive class is a subclass of it. Lemma 1.13 of [Schloeder] p. 2. (Contributed by NM, 4-Oct-2003.)
⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	ordsssuc 6423	An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	onsssuc 6424	A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	ordsssuc2 6425	An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
Theorem	onmindif 6426	When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵))
Theorem	ordnbtwn 6427	There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	onnbtwn 6428	There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. Lemma 1.15 of [Schloeder] p. 2. (Contributed by NM, 9-Jun-1994.)
⊢ (𝐴 ∈ On → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴))
Theorem	sucssel 6429	A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵))
Theorem	orddif 6430	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
Theorem	orduniss 6431	An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
Theorem	ordtri2or 6432	A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	ordtri2or2 6433	A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	ordtri2or3 6434	A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6433. (Contributed by David Moews, 1-May-2017.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
Theorem	ordelinel 6435	The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
Theorem	ordssun 6436	Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶)))
Theorem	ordequn 6437	The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵 ∪ 𝐶) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
Theorem	ordun 6438	The maximum (i.e., union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))
Theorem	onunel 6439	The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
Theorem	ordunisssuc 6440	A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
⊢ ((𝐴 ⊆ On ∧ Ord 𝐵) → (∪ 𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ suc 𝐵))
Theorem	suc11 6441	The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	onun2 6442	The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
Theorem	ontr 6443	An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → Tr 𝐴)
Theorem	onunisuc 6444	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6454. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
Theorem	onordi 6445	An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ Ord 𝐴
Theorem	ontrciOLD 6446	Obsolete version of ontr 6443 as of 28-Dec-2024. (Contributed by NM, 11-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ On    ⇒   ⊢ Tr 𝐴
Theorem	onirri 6447	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ¬ 𝐴 ∈ 𝐴
Theorem	oneli 6448	A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On)
Theorem	onelssi 6449	A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)
Theorem	onssneli 6450	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	onssnel2i 6451	An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ⊆ 𝐴 → ¬ 𝐴 ∈ 𝐵)
Theorem	onelini 6452	An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
Theorem	oneluni 6453	An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
Theorem	onunisuci 6454	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ∪ suc 𝐴 = 𝐴
Theorem	onsseli 6455	Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    ⇒   ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	onun2i 6456	The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
⊢ 𝐴 ∈ On    &   ⊢ 𝐵 ∈ On    ⇒   ⊢ (𝐴 ∪ 𝐵) ∈ On
Theorem	unizlim 6457	An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
Theorem	on0eqel 6458	An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	snsn0non 6459	The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 7846). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 6460. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ ¬ {{∅}} ∈ On
Theorem	onxpdisj 6460	Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6459. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (On ∩ (V × V)) = ∅
Theorem	onnev 6461	The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof shortened by Wolf Lammen, 27-May-2024.)
⊢ On ≠ V
2.3.15  Definite description binder (inverted iota)
Syntax	cio 6462	Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
Theorem	iotajust 6463*	Soundness justification theorem for df-iota 6464. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-iota 6464*	Define Russell's definition description binder, which can be read as "the unique 𝑥 such that 𝜑", where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see iotaval 6482); otherwise, it evaluates to the empty set (see iotanul 6489). Russell used the inverted iota symbol ℩ to represent the binder. Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 7360 (or iotacl 6497 for unbounded iota), as demonstrated in the proof of supub 9410. This can be easier than applying riotasbc 7362 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
Theorem	dfiota2 6465*	Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
Theorem	nfiota1 6466	Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥(℩𝑥𝜑)
Theorem	nfiotadw 6467*	Deduction version of nfiotaw 6468. Version of nfiotad 6469 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiotaw 6468*	Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6470 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	nfiotad 6469	Deduction version of nfiota 6470. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotadw 6467 when possible. (Contributed by NM, 18-Feb-2013.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiota 6470	Bound-variable hypothesis builder for the ℩ class. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfiotaw 6468 when possible. (Contributed by NM, 23-Aug-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	cbviotaw 6471*	Change bound variables in a description binder. Version of cbviota 6473 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Andrew Salmon, 1-Aug-2011.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotavw 6472*	Change bound variables in a description binder. Version of cbviotav 6474 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviota 6473	Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbviotaw 6471 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotav 6474*	Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker cbviotavw 6472 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	sb8iota 6475	Variable substitution in description binder. Compare sb8eu 2593. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Theorem	iotaeq 6476	Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Theorem	iotabi 6477	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Theorem	uniabio 6478*	Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦)
Theorem	iotaval2 6479*	Version of iotaval 6482 using df-iota 6464 instead of dfiota2 6465. (Contributed by SN, 6-Nov-2024.)
⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
Theorem	iotauni2 6480*	Version of iotauni 6486 using df-iota 6464 instead of dfiota2 6465. (Contributed by SN, 6-Nov-2024.)
⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
Theorem	iotanul2 6481*	Version of iotanul 6489 using df-iota 6464 instead of dfiota2 6465. (Contributed by SN, 6-Nov-2024.)
⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
Theorem	iotaval 6482*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 23-Nov-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Theorem	iotassuni 6483	The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
Theorem	iotaex 6484	Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ∈ V
Theorem	iotavalOLD 6485*	Obsolete version of iotaval 6482 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Theorem	iotauni 6486	Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
Theorem	iotaint 6487	Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
Theorem	iota1 6488	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
Theorem	iotanul 6489	Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
Theorem	iotassuniOLD 6490	Obsolete version of iotassuni 6483 as of 23-Dec-2024. (Contributed by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
Theorem	iotaexOLD 6491	Obsolete version of iotaex 6484 as of 23-Dec-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (℩𝑥𝜑) ∈ V
Theorem	iota4 6492	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
Theorem	iota4an 6493	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)
Theorem	iota5 6494*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	iotabidv 6495*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))
Theorem	iotabii 6496	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑥𝜓)
Theorem	iotacl 6497	Membership law for descriptions. This can be useful for expanding an unbounded iota-based definition (see df-iota 6464). If you have a bounded iota-based definition, riotacl2 7360 may be useful. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥 ∣ 𝜑})
Theorem	iota2df 6498	A condition that allows to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → ∃!𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Theorem	iota2d 6499*	A condition that allows to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → ∃!𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))
Theorem	iota2 6500*	The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))