P. 93 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ordfin 9201	A generalization of onfin 9200 to include the class of all ordinals. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (Ord 𝐴 → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))
Theorem	onfin2 9202	A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
⊢ ω = (On ∩ Fin)
Theorem	nndomo 9203	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	nnsdomo 9204	Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≺ 𝐵 ↔ 𝐴 ⊊ 𝐵))
Theorem	sucdom 9205	Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-pow 5338. (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵))
Theorem	snnen2o 9206	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.) Avoid ax-pow 5338, ax-un 7734. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ ¬ {𝐴} ≈ 2o
Theorem	0sdom1dom 9207	Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7734, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7734. (Revised by BTernaryTau, 7-Dec-2024.)
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
Theorem	0sdom1domALT 9208	Alternate proof of 0sdom1dom 9207, shorter but requiring ax-un 7734. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
Theorem	1sdom2 9209	Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7734, see 1sdom2ALT 9210. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7734. (Revised by BTernaryTau, 8-Dec-2024.)
⊢ 1o ≺ 2o
Theorem	1sdom2ALT 9210	Alternate proof of 1sdom2 9209, shorter but requiring ax-un 7734. (Contributed by NM, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 1o ≺ 2o
Theorem	sdom1 9211	A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5338, ax-un 7734. (Revised by BTernaryTau, 12-Dec-2024.)
⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅)
Theorem	modom 9212	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o)
Theorem	modom2 9213*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o)
Theorem	rex2dom 9214*	A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
Theorem	1sdom2dom 9215	Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.)
⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)
Theorem	1sdom 9216*	A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 9028.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7734. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
Theorem	unxpdomlem1 9217*	Lemma for unxpdom 9220. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)    &   ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)    ⇒   ⊢ (𝑧 ∈ (𝑎 ∪ 𝑏) → (𝐹‘𝑧) = if(𝑧 ∈ 𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
Theorem	unxpdomlem2 9218*	Lemma for unxpdom 9220. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)    &   ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)    &   ⊢ (𝜑 → 𝑤 ∈ (𝑎 ∪ 𝑏))    &   ⊢ (𝜑 → ¬ 𝑚 = 𝑛)    &   ⊢ (𝜑 → ¬ 𝑠 = 𝑡)    ⇒   ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑎 ∧ ¬ 𝑤 ∈ 𝑎)) → ¬ (𝐹‘𝑧) = (𝐹‘𝑤))
Theorem	unxpdomlem3 9219*	Lemma for unxpdom 9220. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ 𝐹 = (𝑥 ∈ (𝑎 ∪ 𝑏) ↦ 𝐺)    &   ⊢ 𝐺 = if(𝑥 ∈ 𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)    ⇒   ⊢ ((1o ≺ 𝑎 ∧ 1o ≺ 𝑏) → (𝑎 ∪ 𝑏) ≼ (𝑎 × 𝑏))
Theorem	unxpdom 9220	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.)
⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐵))
Theorem	unxpdom2 9221	Corollary of unxpdom 9220. (Contributed by NM, 16-Sep-2004.)
⊢ ((1o ≺ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐴))
Theorem	sucxpdom 9222	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.)
⊢ (1o ≺ 𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))
Theorem	pssinf 9223	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ∈ Fin)
Theorem	fisseneq 9224	A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≈ 𝐵) → 𝐴 = 𝐵)
Theorem	ominf 9225	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5338. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ ¬ ω ∈ Fin
Theorem	isinf 9226*	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.) Avoid ax-pow 5338. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
Theorem	fineqvlem 9227	Lemma for fineqv 9228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)
Theorem	fineqv 9228	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)
Theorem	xpfir 9229	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))
Theorem	ssfid 9230	A subset of a finite set is finite, deduction version of ssfi 9158. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ Fin)
Theorem	infi 9231	The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin)
Theorem	rabfi 9232*	A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ Fin)
Theorem	finresfin 9233	The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐵) ∈ Fin)
Theorem	f1finf1o 9234	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.) Avoid ax-pow 5338. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
Theorem	nfielex 9235*	If a class is not finite, then it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	en1eqsn 9236	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5338, ax-un 7734. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})
Theorem	en1eqsnbi 9237	A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20262. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴}))
Theorem	dif1ennnALT 9238	Alternate proof of dif1ennn 9148 using ax-pow 5338. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Theorem	enp1ilem 9239	Lemma for uses of enp1i 9240. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 𝑇 = ({𝑥} ∪ 𝑆)    ⇒   ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇))
Theorem	enp1i 9240*	Proof induction for en2 9241 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5338, ax-un 7734. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ Ord 𝑀    &   ⊢ 𝑁 = suc 𝑀    &   ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑)    &   ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓)
Theorem	en2 9241*	A set equinumerous to ordinal 2 is an unordered pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (𝐴 ≈ 2o → ∃𝑥∃𝑦 𝐴 = {𝑥, 𝑦})
Theorem	en3 9242*	A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (𝐴 ≈ 3o → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧})
Theorem	en4 9243*	A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ (𝐴 ≈ 4o → ∃𝑥∃𝑦∃𝑧∃𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))
Theorem	findcard3 9244*	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.) Avoid ax-pow 5338. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 ∈ Fin → (∀𝑥(𝑥 ⊊ 𝑦 → 𝜑) → 𝜒))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	ac6sfi 9245*	A version of ac6s 10469 for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	frfi 9246	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 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 Fr 𝐴)
Theorem	fimax2g 9247*	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 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)
Theorem	fimaxg 9248*	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 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥))
Theorem	fisupg 9249*	Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧)))
Theorem	wofi 9250	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 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
Theorem	ordunifi 9251	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 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)
Theorem	nnunifi 9252	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)
Theorem	unblem1 9253*	Lemma for unbnn 9257. 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 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) ∧ 𝐴 ∈ 𝐵) → ∩ (𝐵 ∖ suc 𝐴) ∈ 𝐵)
Theorem	unblem2 9254*	Lemma for unbnn 9257. 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 ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
Theorem	unblem3 9255*	Lemma for unbnn 9257. 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 ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))
Theorem	unblem4 9256*	Lemma for unbnn 9257. 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 ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴)
Theorem	unbnn 9257*	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 9629 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)
Theorem	unbnn2 9258*	Version of unbnn 9257 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
⊢ ((ω ∈ V ∧ 𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) → 𝐴 ≈ ω)
Theorem	isfinite2 9259	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)
Theorem	nnsdomg 9260	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of Infinity, we include it as part of the antecedent. See nnsdom 9624 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5338. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Theorem	isfiniteg 9261	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))
Theorem	infsdomnn 9262	An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5338. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴)
Theorem	infn0 9263	An infinite set is not empty. For a shorter proof using ax-un 7734, see infn0ALT 9264. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7734. (Revised by BTernaryTau, 8-Jan-2025.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
Theorem	infn0ALT 9264	Shorter proof of infn0 9263 using ax-un 7734. (Contributed by NM, 23-Oct-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
Theorem	fin2inf 9265	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)
Theorem	unfilem1 9266*	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 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))    ⇒   ⊢ ran 𝐹 = ((𝐴 +o 𝐵) ∖ 𝐴)
Theorem	unfilem2 9267*	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 = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))    ⇒   ⊢ 𝐹:𝐵–1-1-onto→((𝐴 +o 𝐵) ∖ 𝐴)
Theorem	unfilem3 9268	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))
Theorem	unfir 9269	If a union is finite, the operands are finite. Converse of unfi 9156. (Contributed by FL, 3-Aug-2009.)
⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
Theorem	unfib 9270	A union is finite if and only if the operands are finite. (Contributed by AV, 10-May-2025.)
⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
Theorem	unfi2 9271	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 9156 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 9265). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ∪ 𝐵) ≺ ω)
Theorem	difinf 9272	An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∖ 𝐵) ∈ Fin)
Theorem	fodomfi 9273	An onto function implies dominance of domain over range, for finite sets. Unlike fodomg 10507 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5338. (Revised by BTernaryTau, 20-Jun-2025.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
Theorem	fofi 9274	If an onto function has a finite domain, its codomain/range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin)
Theorem	f1fi 9275	If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
Theorem	imafi 9276	Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
Theorem	pwfir 9277	If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
Theorem	pwfilem 9278*	Lemma for pwfi 9279. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5338. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Theorem	pwfi 9279	The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5338. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
Theorem	xpfi 9280	The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5338. (Revised by BTernaryTau, 10-Jan-2025.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
Theorem	3xpfi 9281	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)
Theorem	domunfican 9282	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 ∧ 𝐵 ≈ 𝐴) ∧ ((𝐴 ∩ 𝑋) = ∅ ∧ (𝐵 ∩ 𝑌) = ∅)) → ((𝐴 ∪ 𝑋) ≼ (𝐵 ∪ 𝑌) ↔ 𝑋 ≼ 𝑌))
Theorem	infcntss 9283*	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(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
Theorem	prfi 9284	An unordered pair is finite. For a shorter proof using ax-un 7734, see prfiALT 9285. (Contributed by NM, 22-Aug-2008.) Avoid ax-11 2192, ax-un 7734. (Revised by BTernaryTau, 13-Jan-2025.)
⊢ {𝐴, 𝐵} ∈ Fin
Theorem	prfiALT 9285	Shorter proof of prfi 9284 using ax-un 7734. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝐴, 𝐵} ∈ Fin
Theorem	tpfi 9286	An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
⊢ {𝐴, 𝐵, 𝐶} ∈ Fin
Theorem	fiint 9287*	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.) Use a separate setvar for the right-hand side and avoid ax-pow 5338. (Revised by BTernaryTau, 14-Jan-2025.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ ∀𝑧((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin) → ∩ 𝑧 ∈ 𝐴))
Theorem	fodomfir 9288*	There exists a mapping from a finite set onto any nonempty set that it dominates, proved without using the Axiom of Power Sets (unlike fodomr 9117). (Contributed by BTernaryTau, 23-Jun-2025.)
⊢ ((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)
Theorem	fodomfib 9289*	Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 10511 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5338. (Revised by BTernaryTau, 23-Jun-2025.)
⊢ (𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)))
Theorem	fofinf1o 9290	Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	rneqdmfinf1o 9291	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:𝐴–1-1-onto→𝐴)
Theorem	fidomdm 9292	Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)
Theorem	dmfi 9293	The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin)
Theorem	fundmfibi 9294	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))
Theorem	resfnfinfin 9295	The restriction of a function to a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin)
Theorem	residfi 9296	A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)
Theorem	cnvfiALT 9297	Shorter proof of cnvfi 9161 using ax-pow 5338. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
Theorem	rnfi 9298	The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
Theorem	f1dmvrnfibi 9299	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 9300. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
Theorem	f1vrnfibi 9300	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 9299. (Contributed by AV, 10-Jan-2020.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))