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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlimord 5701 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
(Lim 𝐴 → Ord 𝐴)

Theoremlimuni 5702 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
(Lim 𝐴𝐴 = 𝐴)

Theoremlimuni2 5703 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
(Lim 𝐴 → Lim 𝐴)

Theorem0ellim 5704 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
(Lim 𝐴 → ∅ ∈ 𝐴)

Theoremlimelon 5705 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Theoremonn0 5706 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅

Theoremsuceq 5707 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)

Theoremelsuci 5708 Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))

Theoremelsucg 5709 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremelsuc2g 5710 Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremelsuc 5711 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theoremelsuc2 5712 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))

Theoremnfsuc 5713 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
𝑥𝐴       𝑥 suc 𝐴

Theoremelelsuc 5714 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
(𝐴𝐵𝐴 ∈ suc 𝐵)

Theoremsucel 5715* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
(suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))

Theoremsuc0 5716 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
suc ∅ = {∅}

Theoremsucprc 5717 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
𝐴 ∈ V → suc 𝐴 = 𝐴)

Theoremunisuc 5718 A transitive class is equal to the union of its successor. 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)

Theoremsssucid 5719 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

Theoremsucidg 5720 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
(𝐴𝑉𝐴 ∈ suc 𝐴)

Theoremsucid 5721 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

Theoremnsuceq0 5722 No successor is empty. (Contributed by NM, 3-Apr-1995.)
suc 𝐴 ≠ ∅

Theoremeqelsuc 5723 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
𝐴 ∈ V       (𝐴 = 𝐵𝐴 ∈ suc 𝐵)

Theoremiunsuc 5724* 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 = 𝐴𝐵 = 𝐶)        𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)

Theoremsuctr 5725 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)

TheoremsuctrOLD 5726 Obsolete proof of suctr 5725 as of 24-Sep-2021. (Contributed by Alan Sare, 11-Apr-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

Theoremtrsuc 5727 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 𝐵𝐴) → 𝐵𝐴)

Theoremtrsucss 5728 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
(Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))

Theoremordsssuc 5729 A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)
((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonsssuc 5730 A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremordsssuc2 5731 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 𝐴𝐵 ∈ On) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Theoremonmindif 5732 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))

Theoremordnbtwn 5733 There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

TheoremordnbtwnOLD 5734 Obsolete proof of ordnbtwn 5733 as of 24-Sep-2021. (Contributed by NM, 21-Jun-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremonnbtwn 5735 There is no set between an ordinal number and its successor. Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 9-Jun-1994.)
(𝐴 ∈ On → ¬ (𝐴𝐵𝐵 ∈ suc 𝐴))

Theoremsucssel 5736 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
(𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))

Theoremorddif 5737 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Theoremorduniss 5738 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 𝐴𝐴)

Theoremordtri2or 5739 A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or2 5740 A trichotomy law for ordinal classes. (Contributed by NM, 2-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Theoremordtri2or3 5741 A consequence of total ordering for ordinal classes. Similar to ordtri2or2 5740. (Contributed by David Moews, 1-May-2017.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))

Theoremordelinel 5742 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) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

TheoremordelinelOLD 5743 Obsolete proof of ordelinel 5742 as of 24-Sep-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremordssun 5744 Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Theoremordequn 5745 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 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremordun 5746 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 (𝐴𝐵))

Theoremordunisssuc 5747 A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))

Theoremsuc11 5748 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 𝐵𝐴 = 𝐵))

Theoremonordi 5749 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Ord 𝐴

Theoremontrci 5750 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Tr 𝐴

Theoremonirri 5751 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On        ¬ 𝐴𝐴

Theoremoneli 5752 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       (𝐵𝐴𝐵 ∈ On)

Theoremonelssi 5753 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵𝐴)

Theoremonssneli 5754 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐴𝐵 → ¬ 𝐵𝐴)

Theoremonssnel2i 5755 An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → ¬ 𝐴𝐵)

Theoremonelini 5756 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 = (𝐵𝐴))

Theoremoneluni 5757 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → (𝐴𝐵) = 𝐴)

Theoremonunisuci 5758 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On        suc 𝐴 = 𝐴

Theoremonsseli 5759 Subset is equivalent to membership or equality for ordinal numbers. (Contributed by NM, 15-Sep-1995.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))

Theoremonun2i 5760 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵) ∈ On

Theoremunizlim 5761 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)))

Theoremon0eqel 5762 An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.)
(𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))

Theoremsnsn0non 5763 The singleton of the singleton of the empty set is not an ordinal (nor a natural number by omsson 6961). It can be used to represent an "undefined" value for a partial operation on natural or ordinal numbers. See also onxpdisj 5764. (Contributed by NM, 21-May-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
¬ {{∅}} ∈ On

Theoremonxpdisj 5764 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 5763. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(On ∩ (V × V)) = ∅

Theoremonnev 5765 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.)
On ≠ V

2.3.13  Definite description binder (inverted iota)

Syntaxcio 5766 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)

Theoremiotajust 5767* Soundness justification theorem for df-iota 5768. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}

Definitiondf-iota 5768* 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 5779); otherwise, it evaluates to the empty set (see iotanul 5783). 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 6524 (or iotacl 5791 for unbounded iota), as demonstrated in the proof of supub 8248. This can be easier than applying riotasbc 6526 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 ∣ {𝑥𝜑} = {𝑦}}

Theoremdfiota2 5769* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}

Theoremnfiota1 5770 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)

Theoremnfiotad 5771 Deduction version of nfiota 5772. (Contributed by NM, 18-Feb-2013.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))

Theoremnfiota 5772 Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)

Theoremcbviota 5773 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremcbviotav 5774* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremsb8iota 5775 Variable substitution in description binder. Compare sb8eu 2491. (Contributed by NM, 18-Mar-2013.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Theoremiotaeq 5776 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Theoremiotabi 5777 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Theoremuniabio 5778* 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(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)

Theoremiotaval 5779* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
(∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Theoremiotauni 5780 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiotaint 5781 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiota1 5782 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Theoremiotanul 5783 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) = ∅)

Theoremiotassuni 5784 The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
(℩𝑥𝜑) ⊆ {𝑥𝜑}

Theoremiotaex 5785 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.)
(℩𝑥𝜑) ∈ V

Theoremiota4 5786 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)

Theoremiota4an 5787 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)

Theoremiota5 5788* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremiotabidv 5789* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Theoremiotabii 5790 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)

Theoremiotacl 5791 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5768). If you have a bounded iota-based definition, riotacl2 6524 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Theoremiota2df 5792 A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝑥𝐵)       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2d 5793* A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2 5794* The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Theoremsniota 5795 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})

Theoremdfiota4 5796 The operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.)
(℩𝑥𝜑) = if(∃!𝑥𝜑, {𝑥𝜑}, ∅)

Theoremcsbiota 5797* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)

2.3.14  Functions

Syntaxwfun 5798 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴

Syntaxwfn 5799 Extend the definition of a wff to include the function predicate with a domain. (Read: 𝐴 is a function on 𝐵.)
wff 𝐴 Fn 𝐵

Syntaxwf 5800 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 𝐹 maps 𝐴 into 𝐵.)
wff 𝐹:𝐴𝐵

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