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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnsucsssuc 6101 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4262, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4279. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Theoremnntri3or 6102 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremnntri2 6103 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Theoremnntri1 6104 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremnntri3 6105 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))

Theoremnntri2or2 6106 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))

Theoremnndceq 6107 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4366. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)

Theoremnndcel 6108 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)

Theoremnnsseleq 6109 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Theoremnndifsnid 6110 If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3537 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Theoremnnaordi 6111 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) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaord 6112 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) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Theoremnnaordr 6113 Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))

Theoremnnaword 6114 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 6115 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 6116 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 6117 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +𝑜 𝐴))

Theoremnnawordi 6118 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 6119 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) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmord 6120 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) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Theoremnnmword 6121 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 6122 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 6123 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1𝑜 ∈ ω

Theorem2onn 6124 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2𝑜 ∈ ω

Theorem3onn 6125 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3𝑜 ∈ ω

Theorem4onn 6126 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4𝑜 ∈ ω

Theoremnnm1 6127 Multiply an element of ω by 1𝑜. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·𝑜 1𝑜) = 𝐴)

Theoremnnm2 6128 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 6129 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 6130* 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) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))

Theoremnnawordex 6131* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))

Theoremnnm00 6132 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 6133 Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴

Syntaxcec 6134 Extend the definition of a class to include equivalence class.
class [𝐴]𝑅

Syntaxcqs 6135 Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)

Definitiondf-er 6136 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 6137 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6156, ersymb 6150, and ertr 6151. (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 6137* 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) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))

Definitiondf-ec 6138 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 6137). 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 6139. (Contributed by NM, 23-Jul-1995.)
[𝐴]𝑅 = (𝑅 “ {𝐴})

Theoremdfec2 6139* 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.)
(𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})

Theoremecexg 6140 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
(𝑅𝐵 → [𝐴]𝑅 ∈ V)

Theoremecexr 6141 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)

Definitiondf-qs 6142* Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
(𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}

Theoremereq1 6143 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 6144 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝐴 = 𝐵 → (𝑅 Er 𝐴𝑅 Er 𝐵))

Theoremerrel 6145 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → Rel 𝑅)

Theoremerdm 6146 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → dom 𝑅 = 𝐴)

Theoremercl 6147 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴𝑋)

Theoremersym 6148 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑅𝐴)

Theoremercl2 6149 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑋)

Theoremersymb 6150 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)       (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremertr 6151 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)       (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Theoremertrd 6152 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremertr2d 6153 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐶𝑅𝐴)

Theoremertr3d 6154 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐵𝑅𝐴)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremertr4d 6155 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴𝑅𝐶)

Theoremerref 6156 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)    &   (𝜑𝐴𝑋)       (𝜑𝐴𝑅𝐴)

Theoremercnv 6157 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴𝑅 = 𝑅)

Theoremerrn 6158 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 6159 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 6160 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 → (𝐴𝑉𝑅 ∈ V))

Theoremerexb 6161 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 6162* 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)    &   ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)    &   ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)    &   (𝜑 → (𝑥𝐴𝑥𝑅𝑥))       (𝜑𝑅 Er 𝐴)

Theorembrdifun 6163 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))       ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))

Theoremswoer 6164* 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 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))       (𝜑𝑅 Er 𝑋)

Theoremswoord1 6165* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴 < 𝐶𝐵 < 𝐶))

Theoremswoord2 6166* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))    &   ((𝜑 ∧ (𝑦𝑋𝑧𝑋)) → (𝑦 < 𝑧 → ¬ 𝑧 < 𝑦))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥 < 𝑦 → (𝑥 < 𝑧𝑧 < 𝑦)))    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → (𝐶 < 𝐴𝐶 < 𝐵))

Theoremeqerlem 6167* Lemma for eqer 6168. (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 6168* 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 6169 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 6170 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
∅ Er ∅

Theoremeceq1 6171 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)

Theoremeceq1d 6172 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)

Theoremeceq2 6173 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)

Theoremelecg 6174 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Theoremelec 6175 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 6176 Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
(Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Theoremecss 6177 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 6178* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
(𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)

Theoremereldm 6179 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 → (𝐴𝑋𝐵𝑋))

Theoremerth 6180 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)    &   (𝜑𝐴𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremerth2 6181 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)    &   (𝜑𝐵𝑋)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremerthi 6182 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 6183 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
[𝐴] I = {𝐴}

Theoremqseq1 6184 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))

Theoremqseq2 6185 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))

Theoremelqsg 6186* Closed form of elqs 6187. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))

Theoremelqs 6187* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
𝐵 ∈ V       (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅)

Theoremelqsi 6188* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
(𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥𝐴 𝐵 = [𝑥]𝑅)

Theoremecelqsg 6189 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Theoremecelqsi 6190 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝑅 ∈ V       (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Theoremecopqsi 6191 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
𝑅 ∈ V    &   𝑆 = ((𝐴 × 𝐴) / 𝑅)       ((𝐵𝐴𝐶𝐴) → [⟨𝐵, 𝐶⟩]𝑅𝑆)

Theoremqsexg 6192 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝐴𝑉 → (𝐴 / 𝑅) ∈ V)

Theoremqsex 6193 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
𝐴 ∈ V       (𝐴 / 𝑅) ∈ V

Theoremuniqs 6194 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
(𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Theoremqsss 6195 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝐴)       (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Theoremuniqs2 6196 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
(𝜑𝑅 Er 𝐴)    &   (𝜑𝑅𝑉)       (𝜑 (𝐴 / 𝑅) = 𝐴)

Theoremsnec 6197 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐴 ∈ V       {[𝐴]𝑅} = ({𝐴} / 𝑅)

Theoremecqs 6198 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
𝑅 ∈ V       [𝐴]𝑅 = ({𝐴} / 𝑅)

Theoremecid 6199 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
𝐴 ∈ V       [𝐴] E = 𝐴

Theoremecidg 6200 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
(𝐴𝑉 → [𝐴] E = 𝐴)

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