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Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremphp5 8301 Corollary of the Pigeonhole Principle php 8297: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Theoremsnnen2o 8302 A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.)
¬ {𝐴} ≈ 2𝑜

2.4.27  Finite sets

Theoremonomeneq 8303 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Theoremonfin 8304 An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))

Theoremonfin2 8305 A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
ω = (On ∩ Fin)

Theoremnnfi 8306 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)

Theoremnndomo 8307 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))

Theoremnnsdomo 8308 Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))

Theoremsucdom2 8309 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(𝐴𝐵 → suc 𝐴𝐵)

Theoremsucdom 8310 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.)
(𝐴 ∈ ω → (𝐴𝐵 ↔ suc 𝐴𝐵))

Theorem0sdom1dom 8311 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
(∅ ≺ 𝐴 ↔ 1𝑜𝐴)

Theorem1sdom2 8312 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
1𝑜 ≺ 2𝑜

Theoremsdom1 8313 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(𝐴 ≺ 1𝑜𝐴 = ∅)

Theoremmodom 8314 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃*𝑥𝜑 ↔ {𝑥𝜑} ≼ 1𝑜)

Theoremmodom2 8315* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃*𝑥 𝑥𝐴𝐴 ≼ 1𝑜)

Theorem1sdom 8316* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8182.) (Contributed by Mario Carneiro, 12-Jan-2013.)
(𝐴𝑉 → (1𝑜𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))

Theoremunxpdomlem1 8317* Lemma for unxpdom 8320. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)       (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))

Theoremunxpdomlem2 8318* Lemma for unxpdom 8320. (Contributed by Mario Carneiro, 13-Jan-2013.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)    &   (𝜑𝑤 ∈ (𝑎𝑏))    &   (𝜑 → ¬ 𝑚 = 𝑛)    &   (𝜑 → ¬ 𝑠 = 𝑡)       ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))

Theoremunxpdomlem3 8319* Lemma for unxpdom 8320. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)       ((1𝑜𝑎 ∧ 1𝑜𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))

Theoremunxpdom 8320 Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))

Theoremunxpdom2 8321 Corollary of unxpdom 8320. (Contributed by NM, 16-Sep-2004.)
((1𝑜𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))

Theoremsucxpdom 8322 Cartesian product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Theorempssinf 8323 A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
((𝐴𝐵𝐴𝐵) → ¬ 𝐵 ∈ Fin)

Theoremfisseneq 8324 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)

Theoremominf 8325 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
¬ ω ∈ Fin

Theoremisinf 8326* Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.)
𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))

Theoremfineqvlem 8327 Lemma for fineqv 8328. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Theoremfineqv 8328 If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.)
(¬ ω ∈ V ↔ Fin = V)

Theoremenfi 8329 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
(𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))

Theoremenfii 8330 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)

Theorempssnn 8331* A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)

Theoremssnnfi 8332 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremssfi 8333 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. (Contributed by NM, 24-Jun-1998.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremdomfi 8334 A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremxpfir 8335 The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)
(((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))

Theoremssfid 8336 A subset of a finite set is finite, deduction version of ssfi 8333. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ∈ Fin)

Theoreminfi 8337 The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremrabfi 8338* A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝐴 ∈ Fin → {𝑥𝐴𝜑} ∈ Fin)

Theoremfinresfin 8339 The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(𝐸 ∈ Fin → (𝐸𝐵) ∈ Fin)

Theoremf1finf1o 8340 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))

Theorem0fin 8341 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
∅ ∈ Fin

Theoremnfielex 8342* If a class is not finite, it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝐴 ∈ Fin → ∃𝑥 𝑥𝐴)

Theoremen1eqsn 8343 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
((𝐴𝐵𝐵 ≈ 1𝑜) → 𝐵 = {𝐴})

Theoremen1eqsnbi 8344 A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 19449. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
(𝐴𝐵 → (𝐵 ≈ 1𝑜𝐵 = {𝐴}))

Theoremdiffi 8345 If 𝐴 is finite, (𝐴𝐵) is finite. (Contributed by FL, 3-Aug-2009.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremdif1en 8346 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Theoremenp1ilem 8347 Lemma for uses of enp1i 8348. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝑇 = ({𝑥} ∪ 𝑆)       (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))

Theoremenp1i 8348* Proof induction for en2i 8147 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝑀 ∈ ω    &   𝑁 = suc 𝑀    &   ((𝐴 ∖ {𝑥}) ≈ 𝑀𝜑)    &   (𝑥𝐴 → (𝜑𝜓))       (𝐴𝑁 → ∃𝑥𝜓)

Theoremen2 8349* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 2𝑜 → ∃𝑥𝑦 𝐴 = {𝑥, 𝑦})

Theoremen3 8350* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 3𝑜 → ∃𝑥𝑦𝑧 𝐴 = {𝑥, 𝑦, 𝑧})

Theoremen4 8351* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 4𝑜 → ∃𝑥𝑦𝑧𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))

Theoremfindcard 8352* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))    &   (𝑥 = 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2 8353* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2s 8354* Variation of findcard2 8353 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2d 8355* Deduction version of findcard2 8353. (Contributed by SO, 16-Jul-2018.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))    &   (𝜑𝐴 ∈ Fin)       (𝜑𝜂)

Theoremfindcard3 8356* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)
(𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝑦 ∈ Fin → (∀𝑥(𝑥𝑦𝜑) → 𝜒))       (𝐴 ∈ Fin → 𝜏)

Theoremac6sfi 8357* A version of ac6s 9469 for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremfrfi 8358 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Po 𝐴𝐴 ∈ Fin) → 𝑅 Fr 𝐴)

Theoremfimax2g 8359* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremfimaxg 8360* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑅𝑥))

Theoremfisupg 8361* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧)))

Theoremwofi 8362 A total order on a finite set is a well-order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)

Theoremordunifi 8363 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → 𝐴𝐴)

Theoremnnunifi 8364 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((𝑆 ⊆ ω ∧ 𝑆 ∈ Fin) → 𝑆 ∈ ω)

Theoremunblem1 8365* Lemma for unbnn 8369. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)
(((𝐵 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐵 𝑥𝑦) ∧ 𝐴𝐵) → (𝐵 ∖ suc 𝐴) ∈ 𝐵)

Theoremunblem2 8366* Lemma for unbnn 8369. The value of the function 𝐹 belongs to the unbounded set of natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))

Theoremunblem3 8367* Lemma for unbnn 8369. The value of the function 𝐹 is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))

Theoremunblem4 8368* Lemma for unbnn 8369. The function 𝐹 maps the set of natural numbers one-to-one to the set of unbounded natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)
𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)       ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)

Theoremunbnn 8369* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 8717 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremunbnn2 8370* Version of unbnn 8369 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦𝐴 𝑥𝑦) → 𝐴 ≈ ω)

Theoremisfinite2 8371 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
(𝐴 ≺ ω → 𝐴 ∈ Fin)

Theoremnnsdomg 8372 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

Theoremisfiniteg 8373 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
(ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω))

Theoreminfsdomnn 8374 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((ω ≼ 𝐴𝐵 ∈ ω) → 𝐵𝐴)

Theoreminfn0 8375 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
(ω ≼ 𝐴𝐴 ≠ ∅)

Theoremfin2inf 8376 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless ω exists. (Contributed by NM, 13-Nov-2003.)
(𝐴 ≺ ω → ω ∈ V)

Theoremunfilem1 8377* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))       ran 𝐹 = ((𝐴 +𝑜 𝐵) ∖ 𝐴)

Theoremunfilem2 8378* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
𝐴 ∈ ω    &   𝐵 ∈ ω    &   𝐹 = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))       𝐹:𝐵1-1-onto→((𝐴 +𝑜 𝐵) ∖ 𝐴)

Theoremunfilem3 8379 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 ≈ ((𝐴 +𝑜 𝐵) ∖ 𝐴))

Theoremunfi 8380 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)

Theoremunfir 8381 If a union is finite, the operands are finite. Converse of unfi 8380. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))

Theoremunfi2 8382 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 8380 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 8376). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴𝐵) ≺ ω)

Theoremdifinf 8383 An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴𝐵) ∈ Fin)

Theoremxpfi 8384 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Theorem3xpfi 8385 The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
(𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)

Theoremdomunfican 8386 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
(((𝐴 ∈ Fin ∧ 𝐵𝐴) ∧ ((𝐴𝑋) = ∅ ∧ (𝐵𝑌) = ∅)) → ((𝐴𝑋) ≼ (𝐵𝑌) ↔ 𝑋𝑌))

Theoreminfcntss 8387* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
𝐴 ∈ V       (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥 ≈ ω))

Theoremprfi 8388 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
{𝐴, 𝐵} ∈ Fin

Theoremtpfi 8389 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
{𝐴, 𝐵, 𝐶} ∈ Fin

Theoremfiint 8390* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite nonempty subcollection of 𝐴 is in 𝐴." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)
(∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐴))

Theoremfnfi 8391 A version of fnex 6633 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Theoremfodomfi 8392 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 9507 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)

Theoremfodomfib 8393* Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 9511 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
(𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴)))

Theoremfofinf1o 8394 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
((𝐹:𝐴onto𝐵𝐴𝐵𝐵 ∈ Fin) → 𝐹:𝐴1-1-onto𝐵)

Theoremrneqdmfinf1o 8395 Any function from a finite set onto the same set must be a bijection. (Contributed by AV, 5-Jul-2021.)
((𝐴 ∈ Fin ∧ 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴) → 𝐹:𝐴1-1-onto𝐴)

Theoremfidomdm 8396 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐹 ∈ Fin → dom 𝐹𝐹)

Theoremdmfi 8397 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
(𝐴 ∈ Fin → dom 𝐴 ∈ Fin)

Theoremfundmfibi 8398 A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
(Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin))

Theoremresfnfinfin 8399 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
((𝐹 Fn 𝐴𝐵 ∈ Fin) → (𝐹𝐵) ∈ Fin)

Theoremresidfi 8400 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
(( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

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