P. 87 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	naddcllem 8601*	Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
Theorem	naddcl 8602	Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
Theorem	naddov 8603*	The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
Theorem	naddov2 8604*	Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
Theorem	naddov3 8605*	Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) ⊆ 𝑥})
Theorem	naddf 8606	Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ +no :(On × On)⟶On
Theorem	naddcom 8607	Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
Theorem	naddrid 8608	Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
Theorem	naddlid 8609	Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝐴 ∈ On → (∅ +no 𝐴) = 𝐴)
Theorem	naddssim 8610	Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
Theorem	naddelim 8611	Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
Theorem	naddel1 8612	Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
Theorem	naddel2 8613	Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +no 𝐴) ∈ (𝐶 +no 𝐵)))
Theorem	naddss1 8614	Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
Theorem	naddss2 8615	Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +no 𝐴) ⊆ (𝐶 +no 𝐵)))
Theorem	naddword1 8616	Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +no 𝐵))
Theorem	naddword2 8617	Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 15-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +no 𝐴))
Theorem	naddunif 8618*	Uniformity theorem for natural addition. If 𝐴 is the upper bound of 𝑋 and 𝐵 is the upper bound of 𝑌, then (𝐴 +no 𝐵) can be expressed in terms of 𝑋 and 𝑌. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})    &   ⊢ (𝜑 → 𝐵 = ∩ {𝑦 ∈ On ∣ 𝑌 ⊆ 𝑦})    ⇒   ⊢ (𝜑 → (𝐴 +no 𝐵) = ∩ {𝑧 ∈ On ∣ (( +no “ (𝑋 × {𝐵})) ∪ ( +no “ ({𝐴} × 𝑌))) ⊆ 𝑧})
Theorem	naddasslem1 8619*	Lemma for naddass 8621. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})
Theorem	naddasslem2 8620*	Lemma for naddass 8621. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no (𝐵 +no 𝐶)) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥)})
Theorem	naddass 8621	Natural ordinal addition is associative. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))
Theorem	nadd32 8622	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ((𝐴 +no 𝐶) +no 𝐵))
Theorem	nadd4 8623	Rearragement of terms in a quadruple sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 +no 𝐵) +no (𝐶 +no 𝐷)) = ((𝐴 +no 𝐶) +no (𝐵 +no 𝐷)))
Theorem	nadd42 8624	Rearragement of terms in a quadruple sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 +no 𝐵) +no (𝐶 +no 𝐷)) = ((𝐴 +no 𝐶) +no (𝐷 +no 𝐵)))
Theorem	naddel12 8625	Natural addition to both sides of ordinal less-than. (Contributed by Scott Fenton, 7-Feb-2025.)
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷)))
Theorem	naddsuc2 8626	Natural addition with successor. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no suc 𝐵) = suc (𝐴 +no 𝐵))
Theorem	naddoa 8627	Natural addition of a natural is the same as regular addition. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))
Theorem	omnaddcl 8628	The naturals are closed under natural addition. (Contributed by Scott Fenton, 20-Aug-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) ∈ ω)
2.4.26  Equivalence relations and classes
Syntax	wer 8629	Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴
Syntax	cec 8630	Extend the definition of a class to include equivalence class.
class [𝐴]𝑅
Syntax	cqs 8631	Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)
Definition	df-er 8632	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 8633 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 8652, ersymb 8646, and ertr 8647. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
Theorem	dfer2 8633*	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 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
Definition	df-ec 8634	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 8633). 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 8635. (Contributed by NM, 23-Jul-1995.)
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
Theorem	dfec2 8635*	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 = {𝑦 ∣ 𝐴𝑅𝑦})
Theorem	ecexg 8636	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V)
Theorem	ecexr 8637	A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
Definition	df-qs 8638*	Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
Theorem	ereq1 8639	Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))
Theorem	ereq2 8640	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵))
Theorem	errel 8641	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → Rel 𝑅)
Theorem	erdm 8642	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
Theorem	ercl 8643	Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝑋)
Theorem	ersym 8644	An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐴)
Theorem	ercl2 8645	Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑋)
Theorem	ersymb 8646	An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	ertr 8647	An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    ⇒   ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Theorem	ertrd 8648	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ertr2d 8649	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐴)
Theorem	ertr3d 8650	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐵𝑅𝐴)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ertr4d 8651	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	erref 8652	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.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐴)
Theorem	ercnv 8653	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
Theorem	errn 8654	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)
Theorem	erssxp 8655	An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
Theorem	erex 8656	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 ∈ 𝑉 → 𝑅 ∈ V))
Theorem	erexb 8657	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))
Theorem	iserd 8658*	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 ∈ 𝐴 ↔ 𝑥𝑅𝑥))    ⇒   ⊢ (𝜑 → 𝑅 Er 𝐴)
Theorem	iseri 8659*	A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8658, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)
⊢ Rel 𝑅    &   ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)    &   ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)    &   ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)    ⇒   ⊢ 𝑅 Er 𝐴
Theorem	iseriALT 8660*	Alternate proof of iseri 8659, avoiding the usage of mptru 1547 and ⊤ as antecedent by using ax-mp 5 and one of the hypotheses as antecedent. This results, however, in a slightly longer proof. (Contributed by AV, 30-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Rel 𝑅    &   ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)    &   ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)    &   ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)    ⇒   ⊢ 𝑅 Er 𝐴
Theorem	brinxper 8661*	Conditions for a reflexive, symmetric and transitive binary relation to be an equivalence relation over a class 𝑉. (Contributed by AV, 11-Jun-2025.)
⊢ (𝑥 ∈ 𝑉 → 𝑥 ∼ 𝑥)    &   ⊢ (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))    &   ⊢ (𝑥 ∈ 𝑉 → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))    ⇒   ⊢ ( ∼ ∩ (𝑉 × 𝑉)) Er 𝑉
Theorem	brdifun 8662	Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))    ⇒   ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
Theorem	swoer 8663*	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)))    ⇒   ⊢ (𝜑 → 𝑅 Er 𝑋)
Theorem	swoord1 8664*	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)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐶 ↔ 𝐵 < 𝐶))
Theorem	swoord2 8665*	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)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐶 < 𝐴 ↔ 𝐶 < 𝐵))
Theorem	swoso 8666*	If the incomparability relation is equivalent to equality in a subset, then the partial order strictly orders the subset. (Contributed by Mario Carneiro, 30-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)))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ∧ 𝑥𝑅𝑦)) → 𝑥 = 𝑦)    ⇒   ⊢ (𝜑 → < Or 𝑌)
Theorem	eqerlem 8667*	Lemma for eqer 8668. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}    ⇒   ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
Theorem	eqer 8668*	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.) (Proof shortened by AV, 1-May-2021.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}    ⇒   ⊢ 𝑅 Er V
Theorem	ider 8669	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
Theorem	0er 8670	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 1-May-2021.)
⊢ ∅ Er ∅
Theorem	eceq1 8671	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
Theorem	eceq1d 8672	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
Theorem	eceq2 8673	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Theorem	eceq2i 8674	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
Theorem	eceq2d 8675	Equality theorem for the 𝐴-coset and 𝐵-coset of 𝐶, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)
Theorem	elecg 8676	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
Theorem	ecref 8677	All elements are in their own equivalence class. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)
Theorem	elec 8678	Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)
Theorem	relelec 8679	Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
Theorem	elecres 8680	Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.)
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
Theorem	elecreseq 8681	The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.)
⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
Theorem	elecex 8682	Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V))
Theorem	ecss 8683	An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    ⇒   ⊢ (𝜑 → [𝐴]𝑅 ⊆ 𝑋)
Theorem	ecdmn0 8684	A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
Theorem	ereldm 8685	Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
Theorem	erth 8686	Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Theorem	erth2 8687	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.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Theorem	erthi 8688	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.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Theorem	erdisj 8689	Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝑅 Er 𝑋 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))
Theorem	ecidsn 8690	An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
⊢ [𝐴] I = {𝐴}
Theorem	qseq1 8691	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
Theorem	qseq2 8692	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Theorem	qseq2i 8693	Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 / 𝐴) = (𝐶 / 𝐵)
Theorem	qseq1d 8694	Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
Theorem	qseq2d 8695	Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Theorem	qseq12 8696	Equality theorem for quotient set. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 / 𝐶) = (𝐵 / 𝐷))
Theorem	0qs 8697	Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
⊢ (∅ / 𝑅) = ∅
Theorem	elqsg 8698*	Closed form of elqs 8699. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
Theorem	elqs 8699*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)
Theorem	elqsi 8700*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)