P. 100 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1stinl 9901	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 9902	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 9903	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 9904	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 9905*	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 9906*	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 9907*	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 9908*	Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9876 and df-inr 9877, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9877 (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))
2.6.12  Cardinal numbers
Syntax	ccrd 9909	Extend class definition to include the cardinal size function.
class card
Syntax	cale 9910	Extend class definition to include the aleph function.
class ℵ
Syntax	ccf 9911	Extend class definition to include the cofinality function.
class cf
Syntax	wacn 9912	The axiom of choice for limited-length sequences.
class AC 𝐴
Definition	df-card 9913*	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 10518 for its value and cardval2 9965 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 10523. 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 9914	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 10038, alephsuc 10040, and alephlim 10039. 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 9915*	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 10218 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 9916*	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 9917*	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 9918	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 9919*	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 9920	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)
Theorem	ennum 9921	Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
Theorem	finnum 9922	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 9923	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	tskwe 9924*	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 9925	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 9926*	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 9927	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 9928	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 9929*	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10518, 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 9930	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10519, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
Theorem	cardonle 9931	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 9932	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	cardidm 9933	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 9934*	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 9935	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 9936	A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
Theorem	cardnn 9937	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 9938	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 9939	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 9940	If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9941 are meant to replace carden 10523 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.)
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵)
Theorem	carden2b 9941	If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9940 are meant to replace carden 10523 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 9942*	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 9943	A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o)
Theorem	carddomi2 9944	Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10526, 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 9945	A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardlim 9946	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 9947	A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9948 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)
Theorem	cardsdomel 9948	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 9949*	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 9950*	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 9951	Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10526, 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 9952	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 9953*	Lemma for cardprc 9954. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}    ⇒   ⊢ ¬ 𝐴 ∈ V
Theorem	cardprc 9954	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 10533 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9494 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 9955*	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 9956*	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 9957	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 9958	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 9959	The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9958. (Contributed by NM, 7-Nov-2008.)
⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
Theorem	cardom 9960	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 9961	Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10523, 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 9962	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 9963	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 9964	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 9965*	An alternate version of the value of the cardinal number of a set. Compare cardval 10518. This theorem could be used to give a simpler definition of card in place of df-card 9913. 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 9966*	An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9213 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
Theorem	fidomtri 9967	Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	fidomtri2 9968	Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	harsdom 9969	The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴))
Theorem	onsdom 9970*	Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → ∃𝑥 ∈ On 𝐴 ≺ 𝑥)
Theorem	harval2 9971*	An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
Theorem	harsucnn 9972	The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)
Theorem	cardmin2 9973*	The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥})
Theorem	pm54.43lem 9974*	In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9942), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}. Here we show that this is equivalent to 𝐴 ≈ 1o so that we can use the latter more convenient notation in pm54.43 9975. (Contributed by NM, 4-Nov-2013.)
⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})
Theorem	pm54.43 9975	Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2. Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9942), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 9974. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem dju1p1e2 10145 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))
Theorem	enpr2 9976	An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9033. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5327, ax-un 7722. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
Theorem	pr2ne 9977	If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5327, ax-un 7722. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))
Theorem	prdom2 9978	An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)
Theorem	en2eqpr 9979	Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
⊢ ((𝐶 ≈ 2o ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵}))
Theorem	en2eleq 9980	Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
Theorem	en2other2 9981	Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)
Theorem	dif1card 9982	The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
Theorem	leweon 9983*	Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9984, this order is not set-like, as the preimage of ⟨1o, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    ⇒   ⊢ 𝐿 We (On × On)
Theorem	r0weon 9984*	A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    &   ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}    ⇒   ⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Theorem	infxpenlem 9985*	Lemma for infxpen 9986. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}    &   ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}    &   ⊢ 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))    &   ⊢ (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚 ∈ 𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚 ∈ 𝑎 𝑚 ≺ 𝑎)))    &   ⊢ 𝑀 = ((1st ‘𝑤) ∪ (2nd ‘𝑤))    &   ⊢ 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))    ⇒   ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	infxpen 9986	Every infinite ordinal is equinumerous to its Cartesian square. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation 𝑅 is a well-ordering of (On × On) with the additional property that 𝑅-initial segments of (𝑥 × 𝑥) (where 𝑥 is a limit ordinal) are of cardinality at most 𝑥. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	xpomen 9987	The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
⊢ (ω × ω) ≈ ω
Theorem	xpct 9988	The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)
Theorem	infxpidm2 9989	Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10534. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Theorem	infxpenc 9990*	A canonical version of infxpen 9986, by a completely different approach (although it uses infxpen 9986 via xpomen 9987). Using Cantor's normal form, we can show that 𝐴 ↑o 𝐵 respects equinumerosity (oef1o 9655), so that all the steps of (ω↑𝑊) · (ω↑𝑊) ≈ ω↑(2𝑊) ≈ (ω↑2)↑𝑊 ≈ ω↑𝑊 can be verified using bijections to do the ordinal commutations. (The assumption on 𝑁 can be satisfied using cnfcom3c 9663.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ω ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑o 𝑊))    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))    &   ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))    &   ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))    &   ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))    &   ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))    &   ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))    &   ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)    &   ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))    ⇒   ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)
Theorem	infxpenc2lem1 9991*	Lemma for infxpenc2 9994. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    ⇒   ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
Theorem	infxpenc2lem2 9992*	Lemma for infxpenc2 9994. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))    &   ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))    &   ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))    &   ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))    &   ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))    &   ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))    &   ⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ ⟨((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)⟩)    &   ⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))    ⇒   ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	infxpenc2lem3 9993*	Lemma for infxpenc2 9994. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))    &   ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))    &   ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    ⇒   ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	infxpenc2 9994*	Existence form of infxpenc 9990. A "uniform" or "canonical" version of infxpen 9986, asserting the existence of a single function 𝑔 that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
Theorem	iunmapdisj 9995*	The union ∪ 𝑛 ∈ 𝐶(𝐴 ↑m 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑m 𝑛)
Theorem	fseqenlem1 9996*	Lemma for fseqen 9999. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)    &   ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ω) → (𝐺‘𝐶):(𝐴 ↑m 𝐶)–1-1→𝐴)
Theorem	fseqenlem2 9997*	Lemma for fseqen 9999. (Contributed by Mario Carneiro, 17-May-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)    &   ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})    &   ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)    ⇒   ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴))
Theorem	fseqdom 9998*	One half of fseqen 9999. (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
Theorem	fseqen 9999*	A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)
⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴))
Theorem	infpwfidom 10000	The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))