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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwcda1 9001 The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 (𝐴 +𝑐 1𝑜))

Theorempwcdaidm 9002 If the natural numbers inject into 𝐴, then 𝒫 𝐴 is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.)
(ω ≼ 𝐴 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)

Theoremcdalepw 9003 If 𝐴 is idempotent under cardinal sum and 𝐵 is dominated by the power set of 𝐴, then so is the cardinal sum of 𝐴 and 𝐵. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 +𝑐 𝐴) ≈ 𝐴𝐵 ≼ 𝒫 𝐴) → (𝐴 +𝑐 𝐵) ≼ 𝒫 𝐴)

Theoremonacda 9004 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 30-May-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ≈ (𝐴 +𝑐 𝐵))

Theoremcardacda 9005 The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ≈ ((card‘𝐴) +𝑜 (card‘𝐵)))

Theoremcdanum 9006 The cardinal sum of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 +𝑐 𝐵) ∈ dom card)

Theoremunnum 9007 The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵) ∈ dom card)

Theoremnnacda 9008 The cardinal and ordinal sums of finite ordinals are equal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 +𝑐 𝐵)) = (𝐴 +𝑜 𝐵))

Theoremficardun 9009 The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (card‘(𝐴𝐵)) = ((card‘𝐴) +𝑜 (card‘𝐵)))

Theoremficardun2 9010 The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴𝐵)) ⊆ ((card‘𝐴) +𝑜 (card‘𝐵)))

Theorempwsdompw 9011* Lemma for domtriom 9250. This is the equinumerosity version of the algebraic identity Σ𝑘𝑛(2↑𝑘) = (2↑𝑛) − 1. (Contributed by Mario Carneiro, 7-Feb-2013.)
((𝑛 ∈ ω ∧ ∀𝑘 ∈ suc 𝑛(𝐵𝑘) ≈ 𝒫 𝑘) → 𝑘𝑛 (𝐵𝑘) ≺ (𝐵𝑛))

Theoremunctb 9012 The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴𝐵) ≼ ω)

Theoreminfcdaabs 9013 Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 +𝑐 𝐵) ≈ 𝐴)

Theoreminfunabs 9014 An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)

Theoreminfcda 9015 The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 +𝑐 𝐵) ≈ (𝐴𝐵))

Theoreminfdif 9016 The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴𝐵) ≈ 𝐴)

Theoreminfdif2 9017 Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴𝐵) ≼ 𝐵𝐴𝐵))

Theoreminfxpdom 9018 Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴𝐵𝐴) → (𝐴 × 𝐵) ≼ 𝐴)

Theoreminfxpabs 9019 Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)

Theoreminfunsdom1 9020 The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 9021 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)

Theoreminfunsdom 9021 The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.)
(((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)

Theoreminfxp 9022 Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴𝐵))

Theorempwcdadom 9023 A property of dominance over a powerset, and a main lemma for gchac 9488. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
(𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 𝐵) → 𝒫 𝐴𝐵)

Theoreminfpss 9024* Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9120. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))

Theoreminfmap2 9025* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9383 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})

2.6.10  The Ackermann bijection

Theoremackbij2lem1 9026 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))

Theoremackbij1lem1 9027 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐴𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵𝐴))

Theoremackbij1lem2 9028 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵𝐴)))

Theoremackbij1lem3 9029 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))

Theoremackbij1lem4 9030 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 19-Nov-2014.)
(𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin))

Theoremackbij1lem5 9031 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 19-Nov-2014.)
(𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +𝑜 (card‘𝒫 𝐴)))

Theoremackbij1lem6 9032 Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))

Theoremackbij1lem7 9033* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 21-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))

Theoremackbij1lem8 9034* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 19-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))

Theoremackbij1lem9 9035* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 19-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +𝑜 (𝐹𝐵)))

Theoremackbij1lem10 9036* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)⟶ω

Theoremackbij1lem11 9037* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin))

Theoremackbij1lem12 9038* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremackbij1lem13 9039* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐹‘∅) = ∅

Theoremackbij1lem14 9040* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹𝐴))

Theoremackbij1lem15 9041* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑐 ∈ ω ∧ 𝑐𝐴 ∧ ¬ 𝑐𝐵)) → ¬ (𝐹‘(𝐴 ∩ suc 𝑐)) = (𝐹‘(𝐵 ∩ suc 𝑐)))

Theoremackbij1lem16 9042* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) = (𝐹𝐵) → 𝐴 = 𝐵))

Theoremackbij1lem17 9043* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)–1-1→ω

Theoremackbij1lem18 9044* Lemma for ackbij1 9045. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝐴))

Theoremackbij1 9045* The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω

Theoremackbij1b 9046* The Ackermann bijection, part 1b: the bijection from ackbij1 9045 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))       (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))

Theoremackbij2lem2 9047* Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1𝐴)–1-1-onto→(card‘(𝑅1𝐴)))

Theoremackbij2lem3 9048* Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴))

Theoremackbij2lem4 9049* Lemma for ackbij2 9050. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))       (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴))

Theoremackbij2 9050* The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))    &   𝐻 = (rec(𝐺, ∅) “ ω)       𝐻: (𝑅1 “ ω)–1-1-onto→ω

Theoremr1om 9051 The set of hereditarily finite sets is countable. See ackbij2 9050 for an explicit bijection that works without Infinity. See also r1omALT 9583. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝑅1‘ω) ≈ ω

Theoremfictb 9052 A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)
(𝐴𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

2.6.11  Cofinality (without Axiom of Choice)

Theoremcflem 9053* A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.)
(𝐴𝑉 → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))

Theoremcfval 9054* Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number 𝐴 is the cardinality (size) of the smallest unbounded subset 𝑦 of the ordinal number. Unbounded means that for every member of 𝐴, there is a member of 𝑦 that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
(𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})

Theoremcff 9055 Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
cf:On⟶On

Theoremcfub 9056* An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
(cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}

Theoremcflm 9057* Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})

Theoremcf0 9058 Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)
(cf‘∅) = ∅

Theoremcardcf 9059 Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
(card‘(cf‘𝐴)) = (cf‘𝐴)

Theoremcflecard 9060 Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
(cf‘𝐴) ⊆ (card‘𝐴)

Theoremcfle 9061 Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
(cf‘𝐴) ⊆ 𝐴

Theoremcfon 9062 The cofinality of any set is an ordinal (although it only makes sense when 𝐴 is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.)
(cf‘𝐴) ∈ On

Theoremcfeq0 9063 Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
(𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Theoremcfsuc 9064 Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
(𝐴 ∈ On → (cf‘suc 𝐴) = 1𝑜)

Theoremcff1 9065* There is always a map from (cf‘𝐴) to 𝐴 (this is a stronger condition than the definition, which only presupposes a map from some 𝑦 ≈ (cf‘𝐴). (Contributed by Mario Carneiro, 28-Feb-2013.)
(𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1𝐴 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfflb 9066* If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 9065 motivate the picture of (cf‘𝐴) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) ⊆ 𝐵))

Theoremcfval2 9067* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
(𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))

Theoremcoflim 9068* A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)
((Lim 𝐴𝐵𝐴) → ( 𝐵 = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝑦))

Theoremcflim3 9069* Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
𝐴 ∈ V       (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))

Theoremcflim2 9070 The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
𝐴 ∈ V       (Lim 𝐴 ↔ Lim (cf‘𝐴))

Theoremcfom 9071 Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.)
(cf‘ω) = ω

Theoremcfss 9072* There is a cofinal subset of 𝐴 of cardinality (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       (Lim 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ (cf‘𝐴) ∧ 𝑥 = 𝐴))

Theoremcfslb 9073 Any cofinal subset of 𝐴 is at least as large as (cf‘𝐴). (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       ((Lim 𝐴𝐵𝐴 𝐵 = 𝐴) → (cf‘𝐴) ≼ 𝐵)

Theoremcfslbn 9074 Any subset of 𝐴 smaller than its cofinality has union less than 𝐴. (This is the contrapositive to cfslb 9073.) (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       ((Lim 𝐴𝐵𝐴𝐵 ≺ (cf‘𝐴)) → 𝐵𝐴)

Theoremcfslb2n 9075* Any small collection of small subsets of 𝐴 cannot have union 𝐴, where "small" means smaller than the cofinality. This is a stronger version of cfslb 9073. This is a common application of cofinality: under AC, (ℵ‘1) is regular, so it is not a countable union of countable sets. (Contributed by Mario Carneiro, 24-Jun-2013.)
𝐴 ∈ V       ((Lim 𝐴 ∧ ∀𝑥𝐵 (𝑥𝐴𝑥 ≺ (cf‘𝐴))) → (𝐵 ≺ (cf‘𝐴) → 𝐵𝐴))

Theoremcofsmo 9076* Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
𝐶 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}    &   𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}    &   𝑂 = OrdIso( E , 𝐶)       ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))

Theoremcfsmolem 9077* Lemma for cfsmo 9078. (Contributed by Mario Carneiro, 28-Feb-2013.)
𝐹 = (𝑧 ∈ V ↦ ((𝑔‘dom 𝑧) ∪ 𝑡 ∈ dom 𝑧 suc (𝑧𝑡)))    &   𝐺 = (recs(𝐹) ↾ (cf‘𝐴))       (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfsmo 9078* The map in cff1 9065 can be assumed to be a strictly monotone ordinal function without loss of generality. (Contributed by Mario Carneiro, 28-Feb-2013.)
(𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓𝑤)))

Theoremcfcoflem 9079* Lemma for cfcof 9081, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))

Theoremcoftr 9080* If there is a cofinal map from 𝐵 to 𝐴 and another from 𝐶 to 𝐴, then there is also a cofinal map from 𝐶 to 𝐵. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 9081. (Contributed by Mario Carneiro, 16-Mar-2013.)
𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})       (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))

Theoremcfcof 9081* If there is a cofinal map from 𝐴 to 𝐵, then they have the same cofinality. This was used as Definition 11.1 of [TakeutiZaring] p. 100, who defines an equivalence relation cof (𝐴, 𝐵) and defines our cf(𝐵) as the minimum 𝐵 such that cof (𝐴, 𝐵). (Contributed by Mario Carneiro, 20-Mar-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (cf‘𝐴) = (cf‘𝐵)))

Theoremcfidm 9082 The cofinality function is idempotent. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
(cf‘(cf‘𝐴)) = (cf‘𝐴)

Theoremalephsing 9083 The cofinality of a limit aleph is the same as the cofinality of its argument, so if (ℵ‘𝐴) < 𝐴, then (ℵ‘𝐴) is singular. Conversely, if (ℵ‘𝐴) is regular (i.e. weakly inaccessible), then (ℵ‘𝐴) = 𝐴, so 𝐴 has to be rather large (see alephfp 8916). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
(Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))

2.6.12  Eight inequivalent definitions of finite set

Theoremsornom 9084* The range of a single-step monotone function from ω into a partially ordered set is a chain. (Contributed by Stefan O'Rear, 3-Nov-2014.)
((𝐹 Fn ω ∧ ∀𝑎 ∈ ω ((𝐹𝑎)𝑅(𝐹‘suc 𝑎) ∨ (𝐹𝑎) = (𝐹‘suc 𝑎)) ∧ 𝑅 Po ran 𝐹) → 𝑅 Or ran 𝐹)

Syntaxcfin1a 9085 Extend class notation to include the class of Ia-finite sets.
class FinIa

Syntaxcfin2 9086 Extend class notation to include the class of II-finite sets.
class FinII

Syntaxcfin4 9087 Extend class notation to include the class of IV-finite sets.
class FinIV

Syntaxcfin3 9088 Extend class notation to include the class of III-finite sets.
class FinIII

Syntaxcfin5 9089 Extend class notation to include the class of V-finite sets.
class FinV

Syntaxcfin6 9090 Extend class notation to include the class of VI-finite sets.
class FinVI

Syntaxcfin7 9091 Extend class notation to include the class of VII-finite sets.
class FinVII

Definitiondf-fin1a 9092* A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 7944 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}

Definitiondf-fin2 9093* A set is II-finite (Tarski finite) iff every nonempty chain of subsets contains a maximum element. Definition II of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [] Or 𝑦) → 𝑦𝑦)}

Definitiondf-fin4 9094* A set is IV-finite (Dedekind finite) iff it has no equinumerous proper subset. Definition IV of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦𝑥𝑦𝑥)}

Definitiondf-fin3 9095 A set is III-finite (weakly Dedekind finite) iff its power set is Dedekind finite. Definition III of [Levy58] p. 2. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}

Definitiondf-fin5 9096 A set is V-finite iff it behaves finitely under +𝑐. Definition V of [Levy58] p. 3. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}

Definitiondf-fin6 9097 A set is VI-finite iff it behaves finitely under ×. Definition VI of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}

Definitiondf-fin7 9098* A set is VII-finite iff it cannot be infinitely well-ordered. Equivalent to definition VII of [Levy58] p. 4. (Contributed by Stefan O'Rear, 12-Nov-2014.)
FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥𝑦}

Theoremisfin1a 9099* Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
(𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))

Theoremfin1ai 9100 Property of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
((𝐴 ∈ FinIa𝑋𝐴) → (𝑋 ∈ Fin ∨ (𝐴𝑋) ∈ Fin))

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