P. 101 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	setrec2lem1 10001*	Lemma for setrec2 10004. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)
Theorem	dffun3f 10002*	Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑧𝐴    ⇒   ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
Theorem	setrec2lem2 10003*	Lemma for setrec2 10004. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Theorem	setrec2 10004*	This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, Theorems setrec1 9999 and setrec2v 10005 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹. We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7888) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.)
⊢ Ⅎ𝑎𝐹    &   ⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	setrec2v 10005*	Version of setrec2 10004 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by Emmett Weisz, 6-Mar-2021.)
⊢ 𝐵 = setrecs(𝐹)    &   ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
2.6.13  Disjoint union
Syntax	cdju 10006	Extend class notation to include disjoint union of two classes.
class (𝐴 ⊔ 𝐵)
Syntax	cinl 10007	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 10008	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-dju 10009	Disjoint union of two classes. This is a way of creating a set which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
Definition	df-inl 10010	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
Definition	df-inr 10011	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
Theorem	djueq12 10012	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))
Theorem	djueq1 10013	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))
Theorem	djueq2 10014	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
Theorem	nfdju 10015	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)
Theorem	djuex 10016	The disjoint union of sets is a set. For a shorter proof using djuss 10028 see djuexALT 10030. (Contributed by AV, 28-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djuexb 10017	The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djulcl 10018	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurcl 10019	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djulf1o 10020	The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inl:V⟶1-1onto({∅} × V)
Theorem	djurf1o 10021	The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inr:V⟶1-1onto({1o} × V)
Theorem	inlresf 10022	The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
Theorem	inlresf1 10023	The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴⟶1-1(𝐴 ⊔ 𝐵)
Theorem	inrresf 10024	The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵)
Theorem	inrresf1 10025	The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inr ↾ 𝐵):𝐵⟶1-1(𝐴 ⊔ 𝐵)
Theorem	djuin 10026	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Theorem	djur 10027*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
Theorem	djuss 10028	A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	djuunxp 10029	The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.)
⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	djuexALT 10030	Alternate proof of djuex 10016, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	eldju1st 10031	The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.)
⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
Theorem	eldju2ndl 10032	The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴)
Theorem	eldju2ndr 10033	The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵)
Theorem	djuun 10034	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
Theorem	1stinl 10035	The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Theorem	2ndinl 10036	The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Theorem	1stinr 10037	The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)
Theorem	2ndinr 10038	The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Theorem	updjudhf 10039*	The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
Theorem	updjudhcoinlf 10040*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Theorem	updjudhcoinrg 10041*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
Theorem	updjud 10042*	Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 10010 and df-inr 10011, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 10011 (25-Aug-2023). This is a special case of Example 1 of coproducts in Section 10.67 of [Adamek] p. 185. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
2.6.14  Cardinal numbers
Syntax	ccrd 10043	Extend class definition to include the cardinal size function.
class card
Syntax	cale 10044	Extend class definition to include the aleph function.
class ℵ
Syntax	ccf 10045	Extend class definition to include the cofinality function.
class cf
Syntax	wacn 10046	The axiom of choice for limited-length sequences.
class AC 𝐴
Definition	df-card 10047*	Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 10657 for its value and cardval2 10099 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 10662. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
Definition	df-aleph 10048	Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as Theorems aleph0 10172, alephsuc 10174, and alephlim 10173. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.)
⊢ ℵ = rec(har, ω)
Definition	df-cf 10049*	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 10351 for its value and a description. (Contributed by NM, 1-Apr-2004.)
⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))})
Definition	df-acn 10050*	Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 𝐴 is that for all families of nonempty subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶𝒫 𝑋 ∖ {∅}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
Theorem	cardf2 10051*	The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
Theorem	cardon 10052	The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ (card‘𝐴) ∈ On
Theorem	isnum2 10053*	A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
Theorem	isnumi 10054	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)
Theorem	ennum 10055	Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
Theorem	finnum 10056	Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
Theorem	onenon 10057	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	tskwe 10058*	A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Theorem	xpnum 10059	The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Theorem	cardval3 10060*	An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	cardid2 10061	Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
Theorem	isnum3 10062	A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)
Theorem	oncardval 10063*	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10657, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	oncardid 10064	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10658, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
Theorem	cardonle 10065	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
Theorem	card0 10066	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	cardidm 10067	The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (card‘(card‘𝐴)) = (card‘𝐴)
Theorem	oncard 10068*	A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))
Theorem	ficardom 10069	The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
Theorem	ficardid 10070	A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
Theorem	cardnn 10071	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
Theorem	cardnueq0 10072	The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
Theorem	cardne 10073	No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
Theorem	carden2a 10074	If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 10075 are meant to replace carden 10662 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.)
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)
Theorem	carden2b 10075	If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10074 are meant to replace carden 10662 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
Theorem	card1 10076*	A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	cardsn 10077	A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o)
Theorem	carddomi2 10078	Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10665, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵))
Theorem	sdomsdomcardi 10079	A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardlim 10080	An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))
Theorem	cardsdomelir 10081	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 10082 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardsdomel 10082	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵)))
Theorem	iscard 10083*	Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴))
Theorem	iscard2 10084*	Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥)))
Theorem	carddom2 10085	Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10665, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
Theorem	harcard 10086	The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ (card‘(har‘𝐴)) = (har‘𝐴)
Theorem	cardprclem 10087*	Lemma for cardprc 10088. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}    ⇒   ⊢ ¬ 𝐴 ∈ V
Theorem	cardprc 10088	The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 10672 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9565 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V
Theorem	carduni 10089*	The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴))
Theorem	cardiun 10090*	The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))
Theorem	cardennn 10091	If 𝐴 is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ ω) → (card‘𝐴) = 𝐵)
Theorem	cardsucinf 10092	The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘suc 𝐴) = (card‘𝐴))
Theorem	cardsucnn 10093	The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 10092. (Contributed by NM, 7-Nov-2008.)
⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
Theorem	cardom 10094	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)
⊢ (card‘ω) = ω
Theorem	carden2 10095	Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10662, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))
Theorem	cardsdom2 10096	A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵))
Theorem	domtri2 10097	Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	nnsdomel 10098	Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵))
Theorem	cardval2 10099*	An alternate version of the value of the cardinal number of a set. Compare cardval 10657. This theorem could be used to give a simpler definition of card in place of df-card 10047. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
Theorem	isinffi 10100*	An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9284 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵⟶1-1𝐴)