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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrankxplim 8501 The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 8504 for the successor case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
 
Theoremrankxplim2 8502 If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))
 
Theoremrankxplim3 8503 The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 × 𝐵)))
 
Theoremrankxpsuc 8504 The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 8501 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴𝐵)))
 
Theoremtcwf 8505 The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴 (𝑅1 “ On) → (TC‘𝐴) ∈ (𝑅1 “ On))
 
Theoremtcrank 8506 This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴 (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
 
2.6.6  Scott's trick; collection principle; Hilbert's epsilon
 
Theoremscottex 8507* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
 
Theoremscott0 8508* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)
(𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
 
Theoremscottexs 8509* Theorem scheme version of scottex 8507. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
 
Theoremscott0s 8510* Theorem scheme version of scott0 8508. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.)
(∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)
 
Theoremcplem1 8511* Lemma for the Collection Principle cp 8513. (Contributed by NM, 17-Oct-2003.)
𝐶 = {𝑦𝐵 ∣ ∀𝑧𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)}    &   𝐷 = 𝑥𝐴 𝐶       𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝐷) ≠ ∅)
 
Theoremcplem2 8512* -Lemma for the Collection Principle cp 8513. (Contributed by NM, 17-Oct-2003.)
𝐴 ∈ V       𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)
 
Theoremcp 8513* Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 8507 that collapses a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
𝑤𝑥𝑧 (∃𝑦𝜑 → ∃𝑦𝑤 𝜑)
 
Theorembnd 8514* A very strong generalization of the Axiom of Replacement (compare zfrep6 6902), derived from the Collection Principle cp 8513. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)
(∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)
 
Theorembnd2 8515* A variant of the Boundedness Axiom bnd 8514 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))
 
Theoremkardex 8516* The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)
{𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
 
Theoremkarden 8517* If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 9128). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 8516 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}       (𝐶 = 𝐷𝐴𝐵)
 
Theoremhtalem 8518* Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional 𝑅 We 𝐴 antecedent. The element 𝐵 is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐴 ∈ V    &   𝐵 = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)       ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)
 
Theoremhta 8519* A ZFC emulation of Hilbert's transfinite axiom. The set 𝐵 has the properties of Hilbert's epsilon, except that it also depends on a well-ordering 𝑅. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and http://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires 𝑅 We 𝐴 as an antecedent. Class 𝐴 collects the sets of the least rank for which 𝜑(𝑥) is true. Class 𝐵, which emulates the epsilon, is the minimum element in a well-ordering 𝑅 on 𝐴.

If a well-ordering 𝑅 on 𝐴 can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace 𝑅 with a dummy setvar variable, say 𝑤, and attach 𝑤 We 𝐴 as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, 𝐵 (which will have 𝑤 as a free variable) will no longer be present, and we can eliminate 𝑤 We 𝐴 by applying exlimiv 1811 and weth 9076, using scottexs 8509 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 8518. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   𝐵 = (𝑧𝐴𝑤𝐴 ¬ 𝑤𝑅𝑧)       (𝑅 We 𝐴 → (𝜑[𝐵 / 𝑥]𝜑))
 
2.6.7  Cardinal numbers
 
Syntaxccrd 8520 Extend class definition to include the cardinal size function.
class card
 
Syntaxcale 8521 Extend class definition to include the aleph function.
class
 
Syntaxccf 8522 Extend class definition to include the cofinality function.
class cf
 
Syntaxwacn 8523 The axiom of choice for limited-length sequences.
class AC 𝐴
 
Definitiondf-card 8524* 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 9123 for its value, cardval2 8576 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 9128. 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 ∣ 𝑦𝑥})
 
Definitiondf-aleph 8525 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 8648, alephsuc 8650, and alephlim 8649. 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)
 
Definitiondf-cf 8526* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 8828 for its value and a description. (Contributed by NM, 1-Apr-2004.)
cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑣𝑥𝑢𝑧 𝑣𝑢))})
 
Definitiondf-acn 8527* Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋AC 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)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
 
Theoremcardf2 8528* 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 𝑦𝑥}⟶On
 
Theoremcardon 8529 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
 
Theoremisnum2 8530* 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 𝑥𝐴)
 
Theoremisnumi 8531 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
 
Theoremennum 8532 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
 
Theoremfinnum 8533 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ dom card)
 
Theoremonenon 8534 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ On → 𝐴 ∈ dom card)
 
Theoremtskwe 8535* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
 
Theoremxpnum 8536 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)
 
Theoremcardval3 8537* 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 ∣ 𝑥𝐴})
 
Theoremcardid2 8538 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)
 
Theoremisnum3 8539 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)
 
Theoremoncardval 8540* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 9123, 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 ∣ 𝑥𝐴})
 
Theoremoncardid 8541 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 9124, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
 
Theoremcardonle 8542 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)
 
Theoremcard0 8543 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
(card‘∅) = ∅
 
Theoremcardidm 8544 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
(card‘(card‘𝐴)) = (card‘𝐴)
 
Theoremoncard 8545* 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))
 
Theoremficardom 8546 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)
 
Theoremficardid 8547 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
(𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
 
Theoremcardnn 8548 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)
 
Theoremcardnueq0 8549 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
(𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
 
Theoremcardne 8550 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) → ¬ 𝐴𝐵)
 
Theoremcarden2a 8551 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 8552 are meant to replace carden 9128 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
(((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴𝐵)
 
Theoremcarden2b 8552 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8551 are meant to replace carden 9128 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
 
Theoremcard1 8553* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
((card‘𝐴) = 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremcardsn 8554 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
(𝐴𝑉 → (card‘{𝐴}) = 1𝑜)
 
Theoremcarddomi2 8555 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9131, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
 
Theoremsdomsdomcardi 8556 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
(𝐴 ≺ (card‘𝐵) → 𝐴𝐵)
 
Theoremcardlim 8557 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))
 
Theoremcardsdomelir 8558 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 8559 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
 
Theoremcardsdomel 8559 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) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
 
Theoremiscard 8560* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))
 
Theoremiscard2 8561* 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 (𝐴𝑥𝐴𝑥)))
 
Theoremcarddom2 8562 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9131, 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) ↔ 𝐴𝐵))
 
Theoremharcard 8563 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)
 
Theoremcardprclem 8564* Lemma for cardprc 8565. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}        ¬ 𝐴 ∈ V
 
Theoremcardprc 8565 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 9138 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 8208 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
 
Theoremcarduni 8566* The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.)
(𝐴𝑉 → (∀𝑥𝐴 (card‘𝑥) = 𝑥 → (card‘ 𝐴) = 𝐴))
 
Theoremcardiun 8567* The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
(𝐴𝑉 → (∀𝑥𝐴 (card‘𝐵) = 𝐵 → (card‘ 𝑥𝐴 𝐵) = 𝑥𝐴 𝐵))
 
Theoremcardennn 8568 If 𝐴 is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
((𝐴𝐵𝐵 ∈ ω) → (card‘𝐴) = 𝐵)
 
Theoremcardsucinf 8569 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))
 
Theoremcardsucnn 8570 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 8569. (Contributed by NM, 7-Nov-2008.)
(𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
 
Theoremcardom 8571 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
 
Theoremcarden2 8572 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9128, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremcardsdom2 8573 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) ↔ 𝐴𝐵))
 
Theoremdomtri2 8574 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremnnsdomel 8575 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
 
Theoremcardval2 8576* An alternate version of the value of the cardinal number of a set. Compare cardval 9123. This theorem could be used to give us a simpler definition of card in place of df-card 8524. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
(𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
 
Theoremisinffi 8577* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 7934 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)
 
Theoremfidomtri 8578 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 ∧ 𝐵𝑉) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremfidomtri2 8579 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.)
((𝐴𝑉𝐵 ∈ Fin) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremharsdom 8580 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))
 
Theoremonsdom 8581* 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 𝐴𝑥)
 
Theoremharval2 8582* 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 ∣ 𝐴𝑥})
 
Theoremcardmin2 8583* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
(∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
 
Theorempm54.43lem 8584* 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 8553), 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 8585. (Contributed by NM, 4-Nov-2013.)
(𝐴 ≈ 1𝑜𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜})
 
Theorempm54.43 8585 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 8553), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1𝑜} which is the same as 𝐴 ≈ 1𝑜 by pm54.43lem 8584. 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 pm110.643 8758 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

((𝐴 ≈ 1𝑜𝐵 ≈ 1𝑜) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2𝑜))
 
Theorempr2nelem 8586 Lemma for pr2ne 8587. (Contributed by FL, 17-Aug-2008.)
((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2𝑜)
 
Theorempr2ne 8587 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2𝑜𝐴𝐵))
 
Theoremprdom2 8588 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
((𝐴𝐶𝐵𝐷) → {𝐴, 𝐵} ≼ 2𝑜)
 
Theoremen2eqpr 8589 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐶 ≈ 2𝑜𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐶 = {𝐴, 𝐵}))
 
Theoremen2eleq 8590 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.)
((𝑋𝑃𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})
 
Theoremen2other2 8591 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝑋𝑃𝑃 ≈ 2𝑜) → (𝑃 ∖ { (𝑃 ∖ {𝑋})}) = 𝑋)
 
Theoremdif1card 8592 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 ∧ 𝑋𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
 
Theoremleweon 8593* Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 8594, 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)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}       𝐿 We (On × On)
 
Theoremr0weon 8594* 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)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}    &   𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}       (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
 
Theoreminfxpenlem 8595* Lemma for infxpen 8596. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}    &   𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}    &   𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))    &   (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))    &   𝑀 = ((1st𝑤) ∪ (2nd𝑤))    &   𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))       ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
 
Theoreminfxpen 8596 Every infinite ordinal is equinumerous to its Cartesian product. 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)
 
Theoremxpomen 8597 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
 
Theoremxpct 8598 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)
 
Theoreminfxpidm2 8599 The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 9139. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
 
Theoreminfxpenc 8600* A canonical version of infxpen 8596, by a completely different approach (although it uses infxpen 8596 via xpomen 8597). Using Cantor's normal form, we can show that 𝐴𝑜 𝐵 respects equinumerosity (oef1o 8354), 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 8362.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ω ⊆ 𝐴)    &   (𝜑𝑊 ∈ (On ∖ 1𝑜))    &   (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)    &   (𝜑 → (𝐹‘∅) = ∅)    &   (𝜑𝑁:𝐴1-1-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(𝑌𝑋))))    &   𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))    &   𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((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 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)    &   𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))       (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
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