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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnmsuc 6501 Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
 
Theoremnna0r 6502 Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
(𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
 
Theoremnnm0r 6503 Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
(𝐴 ∈ ω → (∅ ·o 𝐴) = ∅)
 
Theoremnnacl 6504 Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
 
Theoremnnmcl 6505 Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
 
Theoremnnacli 6506 ω is closed under addition. Inference form of nnacl 6504. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 +o 𝐵) ∈ ω
 
Theoremnnmcli 6507 ω is closed under multiplication. Inference form of nnmcl 6505. (Contributed by Scott Fenton, 20-Apr-2012.)
𝐴 ∈ ω    &   𝐵 ∈ ω       (𝐴 ·o 𝐵) ∈ ω
 
Theoremnnacom 6508 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 6509 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 6510 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 6511 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 6512 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 6513 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 6514 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 6515 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4525, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4547. (Contributed by Jim Kingdon, 25-Aug-2019.)
(𝐵 ∈ ω → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
 
Theoremnnsucsssuc 6516 Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucsssucr 4526, but the forward direction, for all ordinals, implies excluded middle as seen as onsucsssucexmid 4544. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
 
Theoremnntri3or 6517 Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 
Theoremnntri2 6518 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremnnsucuniel 6519 Given an element 𝐴 of the union of a natural number 𝐵, suc 𝐴 is an element of 𝐵 itself. The reverse direction holds for all ordinals (sucunielr 4527). The forward direction for all ordinals implies excluded middle (ordsucunielexmid 4548). (Contributed by Jim Kingdon, 13-Mar-2022.)
(𝐵 ∈ ω → (𝐴 𝐵 ↔ suc 𝐴𝐵))
 
Theoremnntri1 6520 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremnntri3 6521 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴𝐵 ∧ ¬ 𝐵𝐴)))
 
Theoremnntri2or2 6522 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐵𝐴))
 
Theoremnndceq 6523 Equality of natural numbers is decidable. Theorem 7.2.6 of [HoTT], p. (varies). For the specific case where 𝐵 is zero, see nndceq0 4635. (Contributed by Jim Kingdon, 31-Aug-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴 = 𝐵)
 
Theoremnndcel 6524 Set membership between two natural numbers is decidable. (Contributed by Jim Kingdon, 6-Sep-2019.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → DECID 𝐴𝐵)
 
Theoremnnsseleq 6525 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremnnsssuc 6526 A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 ∈ suc 𝐵))
 
Theoremnntr2 6527 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremdcdifsnid 6528* If we remove a single element from a set with decidable equality then put it back in, we end up with the original set. This strengthens difsnss 3753 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 17-Jun-2022.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 
Theoremfnsnsplitdc 6529* Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))
 
Theoremfunresdfunsndc 6530* Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
((∀𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)
 
Theoremnndifsnid 6531 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 3753 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 6532 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 6533 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 6534 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 6535 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 6536 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 6537 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 6538 Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴))
 
Theoremnnawordi 6539 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 6540 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 6541 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 6542 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 6543 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 6544 One is a natural number. (Contributed by NM, 29-Oct-1995.)
1o ∈ ω
 
Theorem2onn 6545 The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
2o ∈ ω
 
Theorem3onn 6546 The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
3o ∈ ω
 
Theorem4onn 6547 The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
4o ∈ ω
 
Theorem2ssom 6548 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
2o ⊆ ω
 
Theoremnnm1 6549 Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)
 
Theoremnnm2 6550 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 6551 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 6552* 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 6553* 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 6554 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 = ∅)))
 
2.6.25  Equivalence relations and classes
 
Syntaxwer 6555 Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴
 
Syntaxcec 6556 Extend the definition of a class to include equivalence class.
class [𝐴]𝑅
 
Syntaxcqs 6557 Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)
 
Definitiondf-er 6558 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 6559 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6578, ersymb 6572, and ertr 6573. (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 6559* 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 6560 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 6559). 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 6561. (Contributed by NM, 23-Jul-1995.)
[𝐴]𝑅 = (𝑅 “ {𝐴})
 
Theoremdfec2 6561* 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 6562 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
(𝑅𝐵 → [𝐴]𝑅 ∈ V)
 
Theoremecexr 6563 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
 
Definitiondf-qs 6564* 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 6565 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 6566 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝐴 = 𝐵 → (𝑅 Er 𝐴𝑅 Er 𝐵))
 
Theoremerrel 6567 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → Rel 𝑅)
 
Theoremerdm 6568 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
 
Theoremercl 6569 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴𝑋)
 
Theoremersym 6570 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑅𝐴)
 
Theoremercl2 6571 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑋)
 
Theoremersymb 6572 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
(𝜑𝑅 Er 𝑋)       (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 
Theoremertr 6573 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 6574 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremertr2d 6575 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐶𝑅𝐴)
 
Theoremertr3d 6576 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐵𝑅𝐴)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremertr4d 6577 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Er 𝑋)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴𝑅𝐶)
 
Theoremerref 6578 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 6579 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
(𝑅 Er 𝐴𝑅 = 𝑅)
 
Theoremerrn 6580 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 6581 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 6582 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 6583 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 6584* 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 6585 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))       ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremswoer 6586* 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 6587* 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 6588* 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 6589* Lemma for eqer 6590. (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 6590* 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 6591 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 6592 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
∅ Er ∅
 
Theoremeceq1 6593 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
 
Theoremeceq1d 6594 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
 
Theoremeceq2 6595 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
(𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
 
Theoremeceq2i 6596 Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, inference version. (Contributed by Peter Mazsa, 11-May-2021.)
𝐴 = 𝐵       [𝐶]𝐴 = [𝐶]𝐵
 
Theoremeceq2d 6597 Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)
 
Theoremelecg 6598 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
 
Theoremelec 6599 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 6600 Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
(Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
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