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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnmcli 6001 𝜔 is closed under multiplication. Inference form of nnmcl 5999. (Contributed by Scott Fenton, 20-Apr-2012.)
A 𝜔    &   B 𝜔       (A ·𝑜 B) 𝜔
 
Theoremnnacom 6002 Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔) → (A +𝑜 B) = (B +𝑜 A))
 
Theoremnnaass 6003 Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) +𝑜 𝐶) = (A +𝑜 (B +𝑜 𝐶)))
 
Theoremnndi 6004 Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (A ·𝑜 (B +𝑜 𝐶)) = ((A ·𝑜 B) +𝑜 (A ·𝑜 𝐶)))
 
Theoremnnmass 6005 Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶)))
 
Theoremnnmsucr 6006 Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((A 𝜔 B 𝜔) → (suc A ·𝑜 B) = ((A ·𝑜 B) +𝑜 B))
 
Theoremnnmcom 6007 Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((A 𝜔 B 𝜔) → (A ·𝑜 B) = (B ·𝑜 A))
 
Theoremnndir 6008 Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) ·𝑜 𝐶) = ((A ·𝑜 𝐶) +𝑜 (B ·𝑜 𝐶)))
 
Theoremnnsucelsuc 6009 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4199, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4215. (Contributed by Jim Kingdon, 25-Aug-2019.)
(B 𝜔 → (A B ↔ suc A suc B))
 
Theoremnnsucsssuc 6010 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4200, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4212. (Contributed by Jim Kingdon, 25-Aug-2019.)
((A 𝜔 B 𝜔) → (AB ↔ suc A ⊆ suc B))
 
Theoremnntri3or 6011 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((A 𝜔 B 𝜔) → (A B A = B B A))
 
Theoremnntri2 6012 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((A 𝜔 B 𝜔) → (A B ↔ ¬ (A = B B A)))
 
Theoremnntri1 6013 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((A 𝜔 B 𝜔) → (AB ↔ ¬ B A))
 
Theoremnntri3 6014 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((A 𝜔 B 𝜔) → (A = B ↔ (¬ A B ¬ B A)))
 
Theoremnndceq 6015 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where B is zero, see nndceq0 4282. (Contributed by Jim Kingdon, 31-Aug-2019.)
((A 𝜔 B 𝜔) → DECID A = B)
 
Theoremnndcel 6016 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((A 𝜔 B 𝜔) → DECID A B)
 
Theoremnnaordi 6017 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((B 𝜔 𝐶 𝜔) → (A B → (𝐶 +𝑜 A) (𝐶 +𝑜 B)))
 
Theoremnnaord 6018 Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (A B ↔ (𝐶 +𝑜 A) (𝐶 +𝑜 B)))
 
Theoremnnaordr 6019 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (A B ↔ (A +𝑜 𝐶) (B +𝑜 𝐶)))
 
Theoremnnaword 6020 Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (AB ↔ (𝐶 +𝑜 A) ⊆ (𝐶 +𝑜 B)))
 
Theoremnnacan 6021 Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) = (A +𝑜 𝐶) ↔ B = 𝐶))
 
Theoremnnaword1 6022 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔) → A ⊆ (A +𝑜 B))
 
Theoremnnaword2 6023 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((A 𝜔 B 𝜔) → A ⊆ (B +𝑜 A))
 
Theoremnnawordi 6024 Adding to both sides of an inequality in 𝜔 (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.)
((A 𝜔 B 𝜔 𝐶 𝜔) → (AB → (A +𝑜 𝐶) ⊆ (B +𝑜 𝐶)))
 
Theoremnnmordi 6025 Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((B 𝜔 𝐶 𝜔) 𝐶) → (A B → (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
 
Theoremnnmord 6026 Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔 𝐶 𝜔) → ((A B 𝐶) ↔ (𝐶 ·𝑜 A) (𝐶 ·𝑜 B)))
 
Theoremnnmword 6027 Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
(((A 𝜔 B 𝜔 𝐶 𝜔) 𝐶) → (AB ↔ (𝐶 ·𝑜 A) ⊆ (𝐶 ·𝑜 B)))
 
Theoremnnmcan 6028 Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(((A 𝜔 B 𝜔 𝐶 𝜔) A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) ↔ B = 𝐶))
 
Theorem1onn 6029 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1𝑜 𝜔
 
Theorem2onn 6030 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2𝑜 𝜔
 
Theorem3onn 6031 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 𝜔
 
Theorem4onn 6032 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 𝜔
 
Theoremnnm1 6033 Multiply an element of 𝜔 by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.)
(A 𝜔 → (A ·𝑜 1𝑜) = A)
 
Theoremnnm2 6034 Multiply an element of 𝜔 by 2𝑜 (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
(A 𝜔 → (A ·𝑜 2𝑜) = (A +𝑜 A))
 
Theoremnn2m 6035 Multiply an element of 𝜔 by 2𝑜 (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
(A 𝜔 → (2𝑜 ·𝑜 A) = (A +𝑜 A))
 
Theoremnnaordex 6036* Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔) → (A Bx 𝜔 (∅ x (A +𝑜 x) = B)))
 
Theoremnnawordex 6037* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((A 𝜔 B 𝜔) → (ABx 𝜔 (A +𝑜 x) = B))
 
Theoremnnm00 6038 The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)
((A 𝜔 B 𝜔) → ((A ·𝑜 B) = ∅ ↔ (A = ∅ B = ∅)))
 
2.6.24  Equivalence relations and classes
 
Syntaxwer 6039 Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er A
 
Syntaxcec 6040 Extend the definition of a class to include equivalence class.
class [A]𝑅
 
Syntaxcqs 6041 Extend the definition of a class to include quotient set.
class (A / 𝑅)
 
Definitiondf-er 6042 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6043 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6062, ersymb 6056, and ertr 6057. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
(𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
 
Theoremdfer2 6043* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A xyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z))))
 
Definitiondf-ec 6044 Define the 𝑅-coset of A. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of A modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 6043). In this case, A is a representative (member) of the equivalence class [A]𝑅, which contains all sets that are equivalent to A. Definition of [Enderton] p. 57 uses the notation [A] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 6045. (Contributed by NM, 23-Jul-1995.)
[A]𝑅 = (𝑅 “ {A})
 
Theoremdfec2 6045* Alternate definition of 𝑅-coset of A. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → [A]𝑅 = {yA𝑅y})
 
Theoremecexg 6046 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
(𝑅 B → [A]𝑅 V)
 
Theoremecexr 6047 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
(A [B]𝑅B V)
 
Definitiondf-qs 6048* Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
(A / 𝑅) = {yx A y = [x]𝑅}
 
Theoremereq1 6049 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 = 𝑆 → (𝑅 Er A𝑆 Er A))
 
Theoremereq2 6050 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
(A = B → (𝑅 Er A𝑅 Er B))
 
Theoremerrel 6051 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → Rel 𝑅)
 
Theoremerdm 6052 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → dom 𝑅 = A)
 
Theoremercl 6053 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φA 𝑋)
 
Theoremersym 6054 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φB𝑅A)
 
Theoremercl2 6055 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φB 𝑋)
 
Theoremersymb 6056 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → (A𝑅BB𝑅A))
 
Theoremertr 6057 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))
 
Theoremertrd 6058 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φB𝑅𝐶)       (φA𝑅𝐶)
 
Theoremertr2d 6059 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φB𝑅𝐶)       (φ𝐶𝑅A)
 
Theoremertr3d 6060 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φB𝑅A)    &   (φB𝑅𝐶)       (φA𝑅𝐶)
 
Theoremertr4d 6061 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)    &   (φ𝐶𝑅B)       (φA𝑅𝐶)
 
Theoremerref 6062 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φA 𝑋)       (φA𝑅A)
 
Theoremercnv 6063 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A𝑅 = 𝑅)
 
Theoremerrn 6064 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → ran 𝑅 = A)
 
Theoremerssxp 6065 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A𝑅 ⊆ (A × A))
 
Theoremerex 6066 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → (A 𝑉𝑅 V))
 
Theoremerexb 6067 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er A → (𝑅 V ↔ A V))
 
Theoremiserd 6068* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ → Rel 𝑅)    &   ((φ x𝑅y) → y𝑅x)    &   ((φ (x𝑅y y𝑅z)) → x𝑅z)    &   (φ → (x Ax𝑅x))       (φ𝑅 Er A)
 
Theorembrdifun 6069 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))       ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))
 
Theoremswoer 6070* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))       (φ𝑅 Er 𝑋)
 
Theoremswoord1 6071* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))    &   (φB 𝑋)    &   (φ𝐶 𝑋)    &   (φA𝑅B)       (φ → (A < 𝐶B < 𝐶))
 
Theoremswoord2 6072* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((φ (y 𝑋 z 𝑋)) → (y < z → ¬ z < y))    &   ((φ (x 𝑋 y 𝑋 z 𝑋)) → (x < y → (x < z z < y)))    &   (φB 𝑋)    &   (φ𝐶 𝑋)    &   (φA𝑅B)       (φ → (𝐶 < A𝐶 < B))
 
Theoremeqerlem 6073* Lemma for eqer 6074. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
(x = yA = B)    &   𝑅 = {⟨x, y⟩ ∣ A = B}       (z𝑅wz / xA = w / xA)
 
Theoremeqer 6074* Equivalence relation involving equality of dependent classes A(x) and B(y). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
(x = yA = B)    &   𝑅 = {⟨x, y⟩ ∣ A = B}       𝑅 Er V
 
Theoremider 6075 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
I Er V
 
Theorem0er 6076 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
∅ Er ∅
 
Theoremeceq1 6077 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(A = B → [A]𝐶 = [B]𝐶)
 
Theoremeceq1d 6078 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
(φA = B)       (φ → [A]𝐶 = [B]𝐶)
 
Theoremeceq2 6079 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(A = B → [𝐶]A = [𝐶]B)
 
Theoremelecg 6080 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
((A 𝑉 B 𝑊) → (A [B]𝑅B𝑅A))
 
Theoremelec 6081 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
A V    &   B V       (A [B]𝑅B𝑅A)
 
Theoremrelelec 6082 Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
(Rel 𝑅 → (A [B]𝑅B𝑅A))
 
Theoremecss 6083 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)       (φ → [A]𝑅𝑋)
 
Theoremecdmn0m 6084* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
(A dom 𝑅x x [A]𝑅)
 
Theoremereldm 6085 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
(φ𝑅 Er 𝑋)    &   (φ → [A]𝑅 = [B]𝑅)       (φ → (A 𝑋B 𝑋))
 
Theoremerth 6086 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(φ𝑅 Er 𝑋)    &   (φA 𝑋)       (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))
 
Theoremerth2 6087 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
(φ𝑅 Er 𝑋)    &   (φB 𝑋)       (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))
 
Theoremerthi 6088 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
(φ𝑅 Er 𝑋)    &   (φA𝑅B)       (φ → [A]𝑅 = [B]𝑅)
 
Theoremecidsn 6089 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
[A] I = {A}
 
Theoremqseq1 6090 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(A = B → (A / 𝐶) = (B / 𝐶))
 
Theoremqseq2 6091 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(A = B → (𝐶 / A) = (𝐶 / B))
 
Theoremelqsg 6092* Closed form of elqs 6093. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(B 𝑉 → (B (A / 𝑅) ↔ x A B = [x]𝑅))
 
Theoremelqs 6093* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
B V       (B (A / 𝑅) ↔ x A B = [x]𝑅)
 
Theoremelqsi 6094* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
(B (A / 𝑅) → x A B = [x]𝑅)
 
Theoremecelqsg 6095 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑅 𝑉 B A) → [B]𝑅 (A / 𝑅))
 
Theoremecelqsi 6096 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑅 V       (B A → [B]𝑅 (A / 𝑅))
 
Theoremecopqsi 6097 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
𝑅 V    &   𝑆 = ((A × A) / 𝑅)       ((B A 𝐶 A) → [⟨B, 𝐶⟩]𝑅 𝑆)
 
Theoremqsexg 6098 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
(A 𝑉 → (A / 𝑅) V)
 
Theoremqsex 6099 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
A V       (A / 𝑅) V
 
Theoremuniqs 6100 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
(𝑅 𝑉 (A / 𝑅) = (𝑅A))
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