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Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomtr 6201 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
((AB B𝐶) → A𝐶)

Theorementri 6202 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
AB    &   B𝐶       A𝐶

Theorementr2i 6203 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
AB    &   B𝐶       𝐶A

Theorementr3i 6204 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
AB    &   A𝐶       B𝐶

Theorementr4i 6205 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
AB    &   𝐶B       A𝐶

Theoremendomtr 6206 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
((AB B𝐶) → A𝐶)

Theoremdomentr 6207 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
((AB B𝐶) → A𝐶)

Theoremf1imaeng 6208 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐹:A1-1B 𝐶A 𝐶 𝑉) → (𝐹𝐶) ≈ 𝐶)

Theoremf1imaen2g 6209 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6210 does not need ax-setind 4220.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
(((𝐹:A1-1B B 𝑉) (𝐶A 𝐶 𝑉)) → (𝐹𝐶) ≈ 𝐶)

Theoremf1imaen 6210 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
𝐶 V       ((𝐹:A1-1B 𝐶A) → (𝐹𝐶) ≈ 𝐶)

Theoremen0 6211 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
(A ≈ ∅ ↔ A = ∅)

Theoremensn1 6212 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
A V       {A} ≈ 1𝑜

Theoremensn1g 6213 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
(A 𝑉 → {A} ≈ 1𝑜)

Theoremenpr1g 6214 {A, A} has only one element. (Contributed by FL, 15-Feb-2010.)
(A 𝑉 → {A, A} ≈ 1𝑜)

Theoremen1 6215* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
(A ≈ 1𝑜x A = {x})

Theoremen1bg 6216 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
(A 𝑉 → (A ≈ 1𝑜A = { A}))

Theoremreuen1 6217* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!x A φ ↔ {x Aφ} ≈ 1𝑜)

Theoremeuen1 6218 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!xφ ↔ {xφ} ≈ 1𝑜)

Theoremeuen1b 6219* Two ways to express "A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
(A ≈ 1𝑜∃!x x A)

Theoremen1uniel 6220 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(𝑆 ≈ 1𝑜 𝑆 𝑆)

Theorem2dom 6221* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
(2𝑜Ax A y A ¬ x = y)

Theoremfundmen 6222 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐹 V       (Fun 𝐹 → dom 𝐹𝐹)

Theoremfundmeng 6223 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
((𝐹 𝑉 Fun 𝐹) → dom 𝐹𝐹)

Theoremcnven 6224 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
((Rel A A 𝑉) → AA)

Theoremfndmeng 6225 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐹 Fn A A 𝐶) → A𝐹)

Theoremen2sn 6226 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
((A 𝐶 B 𝐷) → {A} ≈ {B})

Theoremsnfig 6227 A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
(A 𝑉 → {A} Fin)

Theoremfiprc 6228 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Fin ∉ V

Theoremunen 6229 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((AB 𝐶𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (A𝐶) ≈ (B𝐷))

Theoremenm 6230* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
((AB x x A) → y y B)

Theoremxpsnen 6231 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
A V    &   B V       (A × {B}) ≈ A

Theoremxpsneng 6232 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
((A 𝑉 B 𝑊) → (A × {B}) ≈ A)

Theoremxp1en 6233 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(A 𝑉 → (A × 1𝑜) ≈ A)

Theoremendisj 6234* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
A V    &   B V       xy((xA yB) (xy) = ∅)

Theoremxpcomf1o 6235* The canonical bijection from (A × B) to (B × A). (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (x (A × B) ↦ {x})       𝐹:(A × B)–1-1-onto→(B × A)

Theoremxpcomco 6236* Composition with the bijection of xpcomf1o 6235 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (x (A × B) ↦ {x})    &   𝐺 = (y B, z A𝐶)       (𝐺𝐹) = (z A, y B𝐶)

Theoremxpcomen 6237 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
A V    &   B V       (A × B) ≈ (B × A)

Theoremxpcomeng 6238 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
((A 𝑉 B 𝑊) → (A × B) ≈ (B × A))

Theoremxpsnen2g 6239 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((A 𝑉 B 𝑊) → ({A} × B) ≈ B)

Theoremxpassen 6240 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
A V    &   B V    &   𝐶 V       ((A × B) × 𝐶) ≈ (A × (B × 𝐶))

Theoremxpdom2 6241 Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐶 V       (AB → (𝐶 × A) ≼ (𝐶 × B))

Theoremxpdom2g 6242 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐶 𝑉 AB) → (𝐶 × A) ≼ (𝐶 × B))

Theoremxpdom1g 6243 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶 𝑉 AB) → (A × 𝐶) ≼ (B × 𝐶))

Theoremxpdom3m 6244* A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
((A 𝑉 B 𝑊 x x B) → A ≼ (A × B))

Theoremxpdom1 6245 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
𝐶 V       (AB → (A × 𝐶) ≼ (B × 𝐶))

Theoremfopwdom 6246 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 V 𝐹:AontoB) → 𝒫 B ≼ 𝒫 A)

Theoremenen1 6247 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(AB → (A𝐶B𝐶))

Theoremenen2 6248 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(AB → (𝐶A𝐶B))

Theoremdomen1 6249 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(AB → (A𝐶B𝐶))

Theoremdomen2 6250 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(AB → (𝐶A𝐶B))

2.6.26  Finite sets

Theoremnnfi 6251 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(A 𝜔 → A Fin)

Theoremenfi 6252 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
(AB → (A Fin ↔ B Fin))

Theoremenfii 6253 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
((B Fin AB) → A Fin)

Theoremssfiexmid 6254* If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
xy((x Fin yx) → y Fin)       (φ ¬ φ)

Theorem0fin 6255 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
Fin

PART 3  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers.

To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 5992 and similar theorems ), going from there to positive integers (df-ni 6288) and then positive rational numbers (df-nqqs 6332) does not involve a major change in approach compared with the Metamath Proof Explorer.

It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero".

3.1  Construction and axiomatization of real and complex numbers

3.1.1  Dedekind-cut construction of real and complex numbers

Syntaxcnpi 6256 The set of positive integers, which is the set of natural numbers 𝜔 with 0 removed.

Note: This is the start of the Dedekind-cut construction of real and _complex numbers.

class N

class +N

Syntaxcmi 6258 Positive integer multiplication.
class ·N

Syntaxclti 6259 Positive integer ordering relation.
class <N

class +pQ

Syntaxcmpq 6261 Positive pre-fraction multiplication.
class ·pQ

Syntaxcltpq 6262 Positive pre-fraction ordering relation.
class <pQ

Syntaxceq 6263 Equivalence class used to construct positive fractions.
class ~Q

Syntaxcnq 6264 Set of positive fractions.
class Q

Syntaxc1q 6265 The positive fraction constant 1.
class 1Q

class +Q

Syntaxcmq 6267 Positive fraction multiplication.
class ·Q

Syntaxcrq 6268 Positive fraction reciprocal operation.
class *Q

Syntaxcltq 6269 Positive fraction ordering relation.
class <Q

Syntaxceq0 6270 Equivalence class used to construct non-negative fractions.
class ~Q0

Syntaxcnq0 6271 Set of non-negative fractions.
class Q0

Syntaxc0q0 6272 The non-negative fraction constant 0.
class 0Q0

class +Q0

Syntaxcmq0 6274 Non-negative fraction multiplication.
class ·Q0

Syntaxcnp 6275 Set of positive reals.
class P

Syntaxc1p 6276 Positive real constant 1.
class 1P

class +P

Syntaxcmp 6278 Positive real multiplication.
class ·P

Syntaxcltp 6279 Positive real ordering relation.
class <P

Syntaxcer 6280 Equivalence class used to construct signed reals.
class ~R

Syntaxcnr 6281 Set of signed reals.
class R

Syntaxc0r 6282 The signed real constant 0.
class 0R

Syntaxc1r 6283 The signed real constant 1.
class 1R

Syntaxcm1r 6284 The signed real constant -1.
class -1R

class +R

Syntaxcmr 6286 Signed real multiplication.
class ·R

Syntaxcltr 6287 Signed real ordering relation.
class <R

Definitiondf-ni 6288 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)
N = (𝜔 ∖ {∅})

Definitiondf-pli 6289 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
+N = ( +𝑜 ↾ (N × N))

Definitiondf-mi 6290 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
·N = ( ·𝑜 ↾ (N × N))

Definitiondf-lti 6291 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)
<N = ( E ∩ (N × N))

Theoremelni 6292 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
(A N ↔ (A 𝜔 A ≠ ∅))

Theorempinn 6293 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
(A NA 𝜔)

Theorempion 6294 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
(A NA On)

Theorempiord 6295 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
(A N → Ord A)

Theoremniex 6296 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
N V

Theorem0npi 6297 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
¬ ∅ N

Theoremelni2 6298 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
(A N ↔ (A 𝜔 A))

Theorem1pi 6299 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
1𝑜 N