P. 92 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	entrfil 9101	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8935). (Contributed by BTernaryTau, 10-Sep-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	enfii 9102	A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5305. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
Theorem	enfi 9103	Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5305, see enfiALT 9104. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5305. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Theorem	enfiALT 9104	Shorter proof of enfi 9103 using ax-pow 5305. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Theorem	domfi 9105	A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin)
Theorem	entrfi 9106	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8935). (Contributed by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	entrfir 9107	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8935). (Contributed by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	domtrfil 9108	Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 8936). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domtrfi 9109	Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 8936). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domtrfir 9110	Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8936). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	f1imaenfi 9111	If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 8943). (Contributed by BTernaryTau, 29-Sep-2024.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)
Theorem	ssdomfi 9112	A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8929). (Contributed by BTernaryTau, 12-Nov-2024.)
⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))
Theorem	ssdomfi2 9113	A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8929). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵)
Theorem	sbthfilem 9114*	Lemma for sbthfi 9115. (Contributed by BTernaryTau, 4-Nov-2024.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
Theorem	sbthfi 9115	Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9017). (Contributed by BTernaryTau, 4-Nov-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
Theorem	domnsymfi 9116	If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 9023). (Contributed by BTernaryTau, 22-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴)
Theorem	sdomdomtrfi 9117	Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 9030). (Contributed by BTernaryTau, 25-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	domsdomtrfi 9118	Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 9032). (Contributed by BTernaryTau, 25-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	sucdom2 9119	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.) Avoid ax-pow 5305. (Revised by BTernaryTau, 4-Dec-2024.)
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)
2.4.33  Pigeonhole Principle
Theorem	phplem1 9120	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Theorem	phplem2 9121	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5305. (Revised by BTernaryTau, 4-Nov-2024.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
Theorem	nneneq 9122	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 11-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	php 9123	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of phplem1 9120, phplem2 9121, nneneq 9122, and this final piece of the proof. (Contributed by NM, 29-May-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 18-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)
Theorem	php2 9124	Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 20-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
Theorem	php3 9125	Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5305. (Revised by BTernaryTau, 26-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
Theorem	php4 9126	Corollary of the Pigeonhole Principle php 9123: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
⊢ (𝐴 ∈ ω → 𝐴 ≺ suc 𝐴)
Theorem	php5 9127	Corollary of the Pigeonhole Principle php 9123: 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)
Theorem	phpeqd 9128	Corollary of the Pigeonhole Principle using equality. Strengthening of php 9123 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow 5305. (Revised by BTernaryTau, 28-Nov-2024.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	nndomog 9129	Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9133 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9133. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5305. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
2.4.34  Finite sets (cont.)
Theorem	onomeneq 9130	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.) Avoid ax-pow 5305. (Revised by BTernaryTau, 2-Dec-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
Theorem	onfin 9131	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))
Theorem	onfin2 9132	A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
⊢ ω = (On ∩ Fin)
Theorem	nndomo 9133	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 9134	Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≺ 𝐵 ↔ 𝐴 ⊊ 𝐵))
Theorem	sucdom 9135	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 5305. (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵))
Theorem	snnen2o 9136	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 5305, ax-un 7674. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ ¬ {𝐴} ≈ 2o
Theorem	0sdom1dom 9137	Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7674, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7674. (Revised by BTernaryTau, 7-Dec-2024.)
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
Theorem	0sdom1domALT 9138	Alternate proof of 0sdom1dom 9137, shorter but requiring ax-un 7674. (Contributed by NM, 28-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
Theorem	1sdom2 9139	Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7674, see 1sdom2ALT 9140. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7674. (Revised by BTernaryTau, 8-Dec-2024.)
⊢ 1o ≺ 2o
Theorem	1sdom2ALT 9140	Alternate proof of 1sdom2 9139, shorter but requiring ax-un 7674. (Contributed by NM, 4-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 1o ≺ 2o
Theorem	sdom1 9141	A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5305, ax-un 7674. (Revised by BTernaryTau, 12-Dec-2024.)
⊢ (𝐴 ≺ 1o ↔ 𝐴 = ∅)
Theorem	modom 9142	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∃*𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≼ 1o)
Theorem	modom2 9143*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o)
Theorem	rex2dom 9144*	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 9145	Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.)
⊢ (1o ≺ 𝐴 ↔ 2o ≼ 𝐴)
Theorem	1sdom 9146*	A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8959.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7674. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
Theorem	unxpdomlem1 9147*	Lemma for unxpdom 9150. (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 9148*	Lemma for unxpdom 9150. (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 9149*	Lemma for unxpdom 9150. (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 9150	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 9151	Corollary of unxpdom 9150. (Contributed by NM, 16-Sep-2004.)
⊢ ((1o ≺ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≼ (𝐴 × 𝐴))
Theorem	sucxpdom 9152	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 9153	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 9154	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 9155	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ ¬ ω ∈ Fin
Theorem	isinf 9156*	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 5305. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
Theorem	fineqvlem 9157	Lemma for fineqv 9158. (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 9158	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 9159	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 9160	A subset of a finite set is finite, deduction version of ssfi 9089. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ Fin)
Theorem	infi 9161	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 9162*	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 9163	The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
⊢ (𝐸 ∈ Fin → (𝐸 ↾ 𝐵) ∈ Fin)
Theorem	f1finf1o 9164	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 5305. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
Theorem	nfielex 9165*	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 9166	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5305, ax-un 7674. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})
Theorem	en1eqsnbi 9167	A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20694. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴}))
Theorem	dif1ennnALT 9168	Alternate proof of dif1ennn 9079 using ax-pow 5305. (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 9169	Lemma for uses of enp1i 9170. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 𝑇 = ({𝑥} ∪ 𝑆)    ⇒   ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇))
Theorem	enp1i 9170*	Proof induction for en2 9171 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5305, ax-un 7674. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ Ord 𝑀    &   ⊢ 𝑁 = suc 𝑀    &   ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑)    &   ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓)
Theorem	en2 9171*	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 9172*	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 9173*	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 9174*	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 5305. (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 9175*	A version of ac6s 10382 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 9176	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 9177*	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 9178*	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 9179*	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 9180	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 9181	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 9182	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 9183*	Lemma for unbnn 9187. 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 9184*	Lemma for unbnn 9187. 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 9185*	Lemma for unbnn 9187. 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 9186*	Lemma for unbnn 9187. 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 9187*	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 9556 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 9188*	Version of unbnn 9187 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 9189	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 9190	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 9551 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5305. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
Theorem	isfiniteg 9191	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 9192	An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5305. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴)
Theorem	infn0 9193	An infinite set is not empty. For a shorter proof using ax-un 7674, see infn0ALT 9194. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7674. (Revised by BTernaryTau, 8-Jan-2025.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
Theorem	infn0ALT 9194	Shorter proof of infn0 9193 using ax-un 7674. (Contributed by NM, 23-Oct-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
Theorem	fin2inf 9195	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 9196*	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 9197*	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 9198	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 9199	If a union is finite, the operands are finite. Converse of unfi 9087. (Contributed by FL, 3-Aug-2009.)
⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
Theorem	unfib 9200	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))