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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfisbth 6901 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
(((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)
 
Theorem0fin 6902 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
∅ ∈ Fin
 
Theoremfin0 6903* A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴))
 
Theoremfin0or 6904* A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))
 
Theoremdiffitest 6905* If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin       𝜑 ∨ ¬ ¬ 𝜑)
 
Theoremfindcard 6906* 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 6907* 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 6908* Variation of findcard2 6907 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 6909* Deduction version of findcard2 6907. If you also need 𝑦 ∈ Fin (which doesn't come for free due to ssfiexmid 6894), use findcard2sd 6910 instead. (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)
 
Theoremfindcard2sd 6910* Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))    &   (𝜑𝐴 ∈ Fin)       (𝜑𝜂)
 
Theoremdiffisn 6911 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin)
 
Theoremdiffifi 6912 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → (𝐴𝐵) ∈ Fin)
 
Theoreminfnfi 6913 An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
(ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin)
 
Theoremominf 6914 The set of natural numbers is not finite. Although we supply this theorem because we can, the more natural way to express "ω is infinite" is ω ≼ ω which is an instance of domrefg 6785. (Contributed by NM, 2-Jun-1998.)
¬ ω ∈ Fin
 
Theoremisinfinf 6915* An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
(ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
 
Theoremac6sfi 6916* Existence of a choice function 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))
 
Theoremtridc 6917* A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)       (𝜑DECID 𝐵𝑅𝐶)
 
Theoremfimax2gtrilemstep 6918* Lemma for fimax2gtri 6919. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑𝑈𝐴)    &   (𝜑𝑍𝐴)    &   (𝜑𝑉𝐴)    &   (𝜑 → ¬ 𝑉𝑈)    &   (𝜑 → ∀𝑦𝑈 ¬ 𝑍𝑅𝑦)       (𝜑 → ∃𝑥𝐴𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
 
Theoremfimax2gtri 6919* A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)
(𝜑𝑅 Po 𝐴)    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremfinexdc 6920* Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 DECID 𝜑) → DECID𝑥𝐴 𝜑)
 
Theoremdfrex2fin 6921* Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 DECID 𝜑) → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))
 
Theoreminfm 6922* An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
(ω ≼ 𝐴 → ∃𝑥 𝑥𝐴)
 
Theoreminfn0 6923 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
(ω ≼ 𝐴𝐴 ≠ ∅)
 
Theoreminffiexmid 6924* If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.)
(𝑥 ∈ Fin ∨ ω ≼ 𝑥)       (𝜑 ∨ ¬ 𝜑)
 
Theoremen2eqpr 6925 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐶 ≈ 2o𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐶 = {𝐴, 𝐵}))
 
Theoremexmidpw 6926 Excluded middle is equivalent to the power set of 1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
(EXMID ↔ 𝒫 1o ≈ 2o)
 
Theoremexmidpweq 6927 Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
(EXMID ↔ 𝒫 1o = 2o)
 
Theorempw1fin 6928 Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
(EXMID ↔ 𝒫 1o ∈ Fin)
 
Theorempw1dc0el 6929 Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
 
Theoremss1o0el1o 6930 Reformulation of ss1o0el1 4212 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.)
(𝐴 ⊆ 1o → (∅ ∈ 𝐴𝐴 = 1o))
 
Theorempw1dc1 6931 If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
(EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
 
Theoremfientri3 6932 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵𝐵𝐴))
 
Theoremnnwetri 6933* A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
(𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
 
Theoremonunsnss 6934 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)
 
Theoremunfiexmid 6935* If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)       (𝜑 ∨ ¬ 𝜑)
 
Theoremunsnfi 6936 Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐵𝑉 ∧ ¬ 𝐵𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin)
 
Theoremunsnfidcex 6937 The 𝐵𝑉 condition in unsnfi 6936. This is intended to show that unsnfi 6936 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
((𝐴 ∈ Fin ∧ ¬ 𝐵𝐴 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ V)
 
Theoremunsnfidcel 6938 The ¬ 𝐵𝐴 condition in unsnfi 6936. This is intended to show that unsnfi 6936 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵𝐴)
 
Theoremunfidisj 6939 The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ Fin)
 
Theoremundifdcss 6940* Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
(𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))
 
Theoremundifdc 6941* Union of complementary parts into whole. This is a case where we can strengthen undifss 3518 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
 
Theoremundiffi 6942 Union of complementary parts into whole. This is a case where we can strengthen undifss 3518 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
 
Theoremunfiin 6943 The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) ∈ Fin) → (𝐴𝐵) ∈ Fin)
 
Theoremprfidisj 6944 A pair is finite if it consists of two unequal sets. For the case where 𝐴 = 𝐵, see snfig 6832. For the cases where one or both is a proper class, see prprc1 3715, prprc2 3716, or prprc 3717. (Contributed by Jim Kingdon, 31-May-2022.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ∈ Fin)
 
Theoremtpfidisj 6945 A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin)
 
Theoremfiintim 6946* If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as 𝑥 and 𝑦 not being equal, or 𝐴 having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

(∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑥((𝑥𝐴𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐴))
 
Theoremxpfi 6947 The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
 
Theorem3xpfi 6948 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)
 
Theoremfisseneq 6949 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)
 
Theoremphpeqd 6950 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6883 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremssfirab 6951* A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑥𝐴 DECID 𝜓)       (𝜑 → {𝑥𝐴𝜓} ∈ Fin)
 
Theoremssfidc 6952* A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
((𝐴 ∈ Fin ∧ 𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵) → 𝐵 ∈ Fin)
 
Theoremsnon0 6953 An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
 
Theoremfnfi 6954 A version of fnex 5754 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 
Theoremfundmfi 6955 The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin)
 
Theoremfundmfibi 6956 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))
 
Theoremresfnfinfinss 6957 The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
((𝐹 Fn 𝐴𝐵 ∈ Fin ∧ 𝐵𝐴) → (𝐹𝐵) ∈ Fin)
 
Theoremrelcnvfi 6958 If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
((Rel 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
 
Theoremfunrnfi 6959 The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
((Rel 𝐴 ∧ Fun 𝐴𝐴 ∈ Fin) → ran 𝐴 ∈ Fin)
 
Theoremf1ofi 6960 If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐵 ∈ Fin)
 
Theoremf1dmvrnfibi 6961 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6962. (Contributed by AV, 10-Jan-2020.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
 
Theoremf1vrnfibi 6962 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6961. (Contributed by AV, 10-Jan-2020.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))
 
Theoremiunfidisj 6963* The finite union of disjoint finite sets is finite. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ Fin ∧ Disj 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 ∈ Fin)
 
Theoremf1finf1o 6964 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))
 
Theoremen1eqsn 6965 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})
 
Theoremen1eqsnbi 6966 A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
(𝐴𝐵 → (𝐵 ≈ 1o𝐵 = {𝐴}))
 
Theoremsnexxph 6967* A case where the antecedent of snexg 4199 is not needed. The class {𝑥𝜑} is from dcextest 4595. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
{{𝑥𝜑}} ∈ V
 
Theorempreimaf1ofi 6968 The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
(𝜑𝐶𝐵)    &   (𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐶 ∈ Fin)       (𝜑 → (𝐹𝐶) ∈ Fin)
 
Theoremfidcenumlemim 6969* Lemma for fidcenum 6973. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
 
Theoremfidcenumlemrks 6970* Lemma for fidcenum 6973. Induction step for fidcenumlemrk 6971. (Contributed by Jim Kingdon, 20-Oct-2022.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐹:𝑁onto𝐴)    &   (𝜑𝐽 ∈ ω)    &   (𝜑 → suc 𝐽𝑁)    &   (𝜑 → (𝑋 ∈ (𝐹𝐽) ∨ ¬ 𝑋 ∈ (𝐹𝐽)))    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
 
Theoremfidcenumlemrk 6971* Lemma for fidcenum 6973. (Contributed by Jim Kingdon, 20-Oct-2022.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐹:𝑁onto𝐴)    &   (𝜑𝐾 ∈ ω)    &   (𝜑𝐾𝑁)    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 ∈ (𝐹𝐾) ∨ ¬ 𝑋 ∈ (𝐹𝐾)))
 
Theoremfidcenumlemr 6972* Lemma for fidcenum 6973. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐹:𝑁onto𝐴)    &   (𝜑𝑁 ∈ ω)       (𝜑𝐴 ∈ Fin)
 
Theoremfidcenum 6973* A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as 𝑛 ∈ ω∃𝑓𝑓:𝑛onto𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝐴 ∈ Fin ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
 
2.6.32  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 6974* Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}        𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
 
Theoremsbthlem2 6975* Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
 
Theoremsbthlemi3 6976* Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
 
Theoremsbthlemi4 6977* Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
 
Theoremsbthlemi5 6978* Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
 
Theoremsbthlemi6 6979* Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
 
Theoremsbthlem7 6980* Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
 
Theoremsbthlemi8 6981* Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
 
Theoremsbthlemi9 6982* Lemma for isbth 6984. (Contributed by NM, 28-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((EXMID𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
 
Theoremsbthlemi10 6983* Lemma for isbth 6984. (Contributed by NM, 28-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))    &   𝐵 ∈ V       ((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)
 
Theoremisbth 6984 Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 𝐴 is smaller (has lower cardinality) than 𝐵 and vice-versa, then 𝐴 and 𝐵 are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 6974 through sbthlemi10 6983; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6983. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 15169. (Contributed by NM, 8-Jun-1998.)
((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)
 
2.6.33  Finite intersections
 
Syntaxcfi 6985 Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
 
Definitiondf-fi 6986* Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 6989). (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})
 
Theoremfival 6987* The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
 
Theoremelfi 6988* Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = 𝑥))
 
Theoremelfi2 6989* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
 
Theoremelfir 6990 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))
 
Theoremssfii 6991 Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐴𝑉𝐴 ⊆ (fi‘𝐴))
 
Theoremfi0 6992 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅
 
Theoremfieq0 6993 A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
 
Theoremfiss 6994 Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉𝐴𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
 
Theoremfiuni 6995 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 𝐴 = (fi‘𝐴))
 
Theoremfipwssg 6996 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝑉𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
 
Theoremfifo 6997* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)       (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
 
Theoremdcfi 6998* Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 DECID 𝜑) → DECID𝑥𝐴 𝜑)
 
2.6.34  Supremum and infimum
 
Syntaxcsup 6999 Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
 
Syntaxcinf 7000 Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
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