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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremen2eqpr 6801 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 6802 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)

Theoremfientri3 6803 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵𝐵𝐴))

Theoremnnwetri 6804* 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 6805 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)

Theoremunfiexmid 6806* 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 6807 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 6808 The 𝐵𝑉 condition in unsnfi 6807. This is intended to show that unsnfi 6807 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 6809 The ¬ 𝐵𝐴 condition in unsnfi 6807. This is intended to show that unsnfi 6807 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 6810 The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ Fin)

Theoremundifdcss 6811* Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
(𝐴 = (𝐵 ∪ (𝐴𝐵)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 DECID 𝑥𝐵))

Theoremundifdc 6812* Union of complementary parts into whole. This is a case where we can strengthen undifss 3443 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))

Theoremundiffi 6813 Union of complementary parts into whole. This is a case where we can strengthen undifss 3443 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))

Theoremunfiin 6814 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 6815 A pair is finite if it consists of two unequal sets. For the case where 𝐴 = 𝐵, see snfig 6708. For the cases where one or both is a proper class, see prprc1 3631, prprc2 3632, or prprc 3633. (Contributed by Jim Kingdon, 31-May-2022.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ∈ Fin)

Theoremtpfidisj 6816 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 6817* 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 6818 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 6819 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 6820 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)

Theoremphpeqd 6821 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6759 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)

Theoremssfirab 6822* 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 6823* 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 6824 An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Theoremfnfi 6825 A version of fnex 5642 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Theoremfundmfi 6826 The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin)

Theoremfundmfibi 6827 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 6828 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 6829 If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
((Rel 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)

Theoremfunrnfi 6830 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 6831 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 6832 A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6833. (Contributed by AV, 10-Jan-2020.)
((𝐴𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremf1vrnfibi 6833 A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6832. (Contributed by AV, 10-Jan-2020.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (𝐹 ∈ Fin ↔ ran 𝐹 ∈ Fin))

Theoremiunfidisj 6834* 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 6835 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 6836 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
((𝐴𝐵𝐵 ≈ 1o) → 𝐵 = {𝐴})

Theoremen1eqsnbi 6837 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 6838* A case where the antecedent of snexg 4108 is not needed. The class {𝑥𝜑} is from dcextest 4495. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
{{𝑥𝜑}} ∈ V

Theorempreimaf1ofi 6839 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 6840* Lemma for fidcenum 6844. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
(𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))

Theoremfidcenumlemrks 6841* Lemma for fidcenum 6844. Induction step for fidcenumlemrk 6842. (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 6842* Lemma for fidcenum 6844. (Contributed by Jim Kingdon, 20-Oct-2022.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐹:𝑁onto𝐴)    &   (𝜑𝐾 ∈ ω)    &   (𝜑𝐾𝑁)    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 ∈ (𝐹𝐾) ∨ ¬ 𝑋 ∈ (𝐹𝐾)))

Theoremfidcenumlemr 6843* Lemma for fidcenum 6844. 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 6844* 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))

2.6.31  Schroeder-Bernstein Theorem

Theoremsbthlem1 6845* Lemma for isbth 6855. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}        𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))

Theoremsbthlem2 6846* Lemma for isbth 6855. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)

Theoremsbthlemi3 6847* Lemma for isbth 6855. (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 6848* Lemma for isbth 6855. (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 6849* Lemma for isbth 6855. (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 6850* Lemma for isbth 6855. (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 6851* Lemma for isbth 6855. (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 6852* Lemma for isbth 6855. (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 6853* Lemma for isbth 6855. (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 6854* Lemma for isbth 6855. (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 6855 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 6845 through sbthlemi10 6854; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6854. 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 13279. (Contributed by NM, 8-Jun-1998.)
((EXMID ∧ (𝐴𝐵𝐵𝐴)) → 𝐴𝐵)

2.6.32  Finite intersections

Syntaxcfi 6856 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 6857* 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 6860). (Contributed by FL, 27-Apr-2008.)
fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = 𝑦})

Theoremfival 6858* 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 6859* 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 6860* 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 6861 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))

Theoremssfii 6862 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 6863 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅

Theoremfieq0 6864 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 6865 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 6866 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 6867 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 6868* 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))

2.6.33  Supremum and infimum

Syntaxcsup 6869 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 6870 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)

Definitiondf-sup 6871* Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. (Contributed by NM, 22-May-1999.)
sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}

Definitiondf-inf 6872 Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)

Theoremsupeq1 6873 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Theoremsupeq1d 6874 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Theoremsupeq1i 6875 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)

Theoremsupeq2 6876 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))

Theoremsupeq3 6877 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Theoremsupeq123d 6878 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Theoremnfsup 6879 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥sup(𝐴, 𝐵, 𝑅)

Theoremsupmoti 6880* Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 7856) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))

Theoremsupeuti 6881* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))

Theoremsupval2ti 6882* Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))

Theoremeqsupti 6883* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))

Theoremeqsuptid 6884* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)    &   ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Theoremsupclti 6885* A supremum belongs to its base class (closure law). See also supubti 6886 and suplubti 6887. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Theoremsupubti 6886* A supremum is an upper bound. See also supclti 6885 and suplubti 6887.

This proof demonstrates how to expand an iota-based definition (df-iota 5088) using riotacl2 5743.

(Contributed by Jim Kingdon, 24-Nov-2021.)

((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))

Theoremsuplubti 6887* A supremum is the least upper bound. See also supclti 6885 and supubti 6886. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))

Theoremsuplub2ti 6888* Bidirectional form of suplubti 6887. (Contributed by Jim Kingdon, 17-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))

Theoremsupelti 6889* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐶 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝐶𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

Theoremsup00 6890 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅

Theoremsupmaxti 6891* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Theoremsupsnti 6892* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoremisotilem 6893* Lemma for isoti 6894. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))

Theoremisoti 6894* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))

Theoremsupisolem 6895* Lemma for supisoti 6897. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)       ((𝜑𝐷𝐴) → ((∀𝑦𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐷 → ∃𝑧𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝐷)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝐷) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))

Theoremsupisoex 6896* Lemma for supisoti 6897. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))

Theoremsupisoti 6897* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Theoreminfeq1 6898 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1d 6899 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1i 6900 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

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