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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fisbth 6901 | Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.) |
⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵) | ||
Theorem | 0fin 6902 | The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
⊢ ∅ ∈ Fin | ||
Theorem | fin0 6903* | A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.) |
⊢ (𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | fin0or 6904* | A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.) |
⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴)) | ||
Theorem | diffitest 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 → (𝐴 ∖ 𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.) |
⊢ ∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin ⇒ ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑) | ||
Theorem | findcard 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 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2 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 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2s 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 = ∅ → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏) | ||
Theorem | findcard2d 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 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | findcard2sd 6910* | Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.) |
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | diffisn 6911 | Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin) | ||
Theorem | diffifi 6912 | Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin) | ||
Theorem | infnfi 6913 | An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.) |
⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin) | ||
Theorem | ominf 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 | ||
Theorem | isinfinf 6915* | An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.) |
⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) | ||
Theorem | ac6sfi 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 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | tridc 6917* | A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.) |
⊢ (𝜑 → 𝑅 Po 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → DECID 𝐵𝑅𝐶) | ||
Theorem | fimax2gtrilemstep 6918* | Lemma for fimax2gtri 6919. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.) |
⊢ (𝜑 → 𝑅 Po 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ (𝜑 → 𝑈 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑉 ∈ 𝐴) & ⊢ (𝜑 → ¬ 𝑉 ∈ 𝑈) & ⊢ (𝜑 → ∀𝑦 ∈ 𝑈 ¬ 𝑍𝑅𝑦) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦) | ||
Theorem | fimax2gtri 6919* | A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.) |
⊢ (𝜑 → 𝑅 Po 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | ||
Theorem | finexdc 6920* | Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∃𝑥 ∈ 𝐴 𝜑) | ||
Theorem | dfrex2fin 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 ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | ||
Theorem | infm 6922* | An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.) |
⊢ (ω ≼ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) | ||
Theorem | infn0 6923 | An infinite set is not empty. (Contributed by NM, 23-Oct-2004.) |
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | ||
Theorem | inffiexmid 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) | ||
Theorem | en2eqpr 6925 | Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐶 ≈ 2o ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵})) | ||
Theorem | exmidpw 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) | ||
Theorem | exmidpweq 6927 | Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.) |
⊢ (EXMID ↔ 𝒫 1o = 2o) | ||
Theorem | pw1fin 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) | ||
Theorem | pw1dc0el 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) | ||
Theorem | ss1o0el1o 6930 | Reformulation of ss1o0el1 4212 using 1o instead of {∅}. (Contributed by BJ, 9-Aug-2024.) |
⊢ (𝐴 ⊆ 1o → (∅ ∈ 𝐴 ↔ 𝐴 = 1o)) | ||
Theorem | pw1dc1 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) | ||
Theorem | fientri3 6932 | Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) | ||
Theorem | nnwetri 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | ||
Theorem | onunsnss 6934 | Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴) | ||
Theorem | unfiexmid 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) → (𝑥 ∪ 𝑦) ∈ Fin) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | unsnfi 6936 | Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∪ {𝐵}) ∈ Fin) | ||
Theorem | unsnfidcex 6937 | The 𝐵 ∈ 𝑉 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 ∧ ¬ 𝐵 ∈ 𝐴 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ V) | ||
Theorem | unsnfidcel 6938 | The ¬ 𝐵 ∈ 𝐴 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 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ Fin) → DECID ¬ 𝐵 ∈ 𝐴) | ||
Theorem | unfidisj 6939 | The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin) | ||
Theorem | undifdcss 6940* | Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.) |
⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵)) | ||
Theorem | undifdc 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.) |
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) | ||
Theorem | undiffi 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 ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) | ||
Theorem | unfiin 6943 | The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | ||
Theorem | prfidisj 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | ||
Theorem | tpfidisj 6945 | A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) | ||
Theorem | fiintim 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.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐴)) | ||
Theorem | xpfi 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) | ||
Theorem | 3xpfi 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) | ||
Theorem | fisseneq 6949 | 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 | phpeqd 6950 | Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6883 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | ssfirab 6951* | A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin) | ||
Theorem | ssfidc 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 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin) | ||
Theorem | snon0 6953 | An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅) | ||
Theorem | fnfi 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 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) | ||
Theorem | fundmfi 6955 | The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.) |
⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin) | ||
Theorem | fundmfibi 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)) | ||
Theorem | resfnfinfinss 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 𝐴 ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) ∈ Fin) | ||
Theorem | relcnvfi 6958 | If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.) |
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) | ||
Theorem | funrnfi 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 ◡𝐴 ∧ 𝐴 ∈ Fin) → ran 𝐴 ∈ Fin) | ||
Theorem | f1ofi 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:𝐴–1-1-onto→𝐵) → 𝐵 ∈ Fin) | ||
Theorem | f1dmvrnfibi 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin)) | ||
Theorem | f1vrnfibi 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.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin)) | ||
Theorem | iunfidisj 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 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin ∧ Disj 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin) | ||
Theorem | f1finf1o 6964 | Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | ||
Theorem | en1eqsn 6965 | A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | ||
Theorem | en1eqsnbi 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.) |
⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) | ||
Theorem | snexxph 6967* | A case where the antecedent of snexg 4199 is not needed. The class {𝑥 ∣ 𝜑} is from dcextest 4595. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.) |
⊢ {{𝑥 ∣ 𝜑}} ∈ V | ||
Theorem | preimaf1ofi 6968 | The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.) |
⊢ (𝜑 → 𝐶 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → 𝐶 ∈ Fin) ⇒ ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin) | ||
Theorem | fidcenumlemim 6969* | Lemma for fidcenum 6973. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.) |
⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴)) | ||
Theorem | fidcenumlemrks 6970* | Lemma for fidcenum 6973. Induction step for fidcenumlemrk 6971. (Contributed by Jim Kingdon, 20-Oct-2022.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) & ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴) & ⊢ (𝜑 → 𝐽 ∈ ω) & ⊢ (𝜑 → suc 𝐽 ⊆ 𝑁) & ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐽))) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))) | ||
Theorem | fidcenumlemrk 6971* | Lemma for fidcenum 6973. (Contributed by Jim Kingdon, 20-Oct-2022.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) & ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴) & ⊢ (𝜑 → 𝐾 ∈ ω) & ⊢ (𝜑 → 𝐾 ⊆ 𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾))) | ||
Theorem | fidcenumlemr 6972* | Lemma for fidcenum 6973. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) & ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴) & ⊢ (𝜑 → 𝑁 ∈ ω) ⇒ ⊢ (𝜑 → 𝐴 ∈ Fin) | ||
Theorem | fidcenum 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:𝑛–onto→𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.) |
⊢ (𝐴 ∈ Fin ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴)) | ||
Theorem | sbthlem1 6974* | Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ ∪ 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) | ||
Theorem | sbthlem2 6975* | Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ (ran 𝑔 ⊆ 𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) ⊆ ∪ 𝐷) | ||
Theorem | sbthlemi3 6976* | Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷)) | ||
Theorem | sbthlemi4 6977* | Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} ⇒ ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) | ||
Theorem | sbthlemi5 6978* | Lemma for isbth 6984. (Contributed by NM, 22-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔 ⊆ 𝐴)) → dom 𝐻 = 𝐴) | ||
Theorem | sbthlemi6 6979* | Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ (((EXMID ∧ ran 𝑓 ⊆ 𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵) | ||
Theorem | sbthlem7 6980* | Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((Fun 𝑓 ∧ Fun ◡𝑔) → Fun 𝐻) | ||
Theorem | sbthlemi8 6981* | Lemma for isbth 6984. (Contributed by NM, 27-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻) | ||
Theorem | sbthlemi9 6982* | Lemma for isbth 6984. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ⇒ ⊢ ((EXMID ∧ 𝑓:𝐴–1-1→𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → 𝐻:𝐴–1-1-onto→𝐵) | ||
Theorem | sbthlemi10 6983* | Lemma for isbth 6984. (Contributed by NM, 28-Mar-1998.) |
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((EXMID ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵) | ||
Theorem | isbth 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 ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵) | ||
Syntax | cfi 6985 | Extend class notation with the function whose value is the class of finite intersections of the elements of a given set. |
class fi | ||
Definition | df-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}) | ||
Theorem | fival 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.) |
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥}) | ||
Theorem | elfi 6988* | Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥)) | ||
Theorem | elfi2 6989* | The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥)) | ||
Theorem | elfir 6990 | Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵)) | ||
Theorem | ssfii 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.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴)) | ||
Theorem | fi0 6992 | The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (fi‘∅) = ∅ | ||
Theorem | fieq0 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅)) | ||
Theorem | fiss 6994 | Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) | ||
Theorem | fiuni 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.) |
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴)) | ||
Theorem | fipwssg 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋) | ||
Theorem | fifo 6997* | Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) | ||
Theorem | dcfi 6998* | Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.) |
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 DECID 𝜑) → DECID ∀𝑥 ∈ 𝐴 𝜑) | ||
Syntax | csup 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) | ||
Syntax | cinf 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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