P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnaordex 6801*	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 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Theorem	nnawordex 6802*	Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵))
Theorem	nnm00 6803	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 = ∅)))
2.6.26  Equivalence relations and classes
Syntax	wer 6804	Extend the definition of a wff to include the equivalence predicate.
wff 𝑅 Er 𝐴
Syntax	cec 6805	Extend the definition of a class to include equivalence class.
class [𝐴]𝑅
Syntax	cqs 6806	Extend the definition of a class to include quotient set.
class (𝐴 / 𝑅)
Definition	df-er 6807	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 6808 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6827, ersymb 6821, and ertr 6822. (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 6808*	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 6809	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 6808). 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 6810. (Contributed by NM, 23-Jul-1995.)
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
Theorem	dfec2 6810*	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 6811	An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V)
Theorem	ecexr 6812	An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)
Definition	df-qs 6813*	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 6814	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 6815	Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵))
Theorem	errel 6816	An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → Rel 𝑅)
Theorem	erdm 6817	The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
Theorem	ercl 6818	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 6819	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 6820	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 6821	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 6822	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 6823	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ertr2d 6824	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐴)
Theorem	ertr3d 6825	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐵𝑅𝐴)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ertr4d 6826	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	erref 6827	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 6828	The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
Theorem	errn 6829	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 6830	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 6831	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 6832	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 6833*	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	brdifun 6834	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 6835*	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 6836*	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 6837*	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	eqerlem 6838*	Lemma for eqer 6839. (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 6839*	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 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}    ⇒   ⊢ 𝑅 Er V
Theorem	ider 6840	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 6841	The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ ∅ Er ∅
Theorem	eceq1 6842	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶)
Theorem	eceq1d 6843	Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
Theorem	eceq2 6844	Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
Theorem	eceq2i 6845	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 6846	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 6847	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	elec 6848	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 6849	Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
Theorem	ecss 6850	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	ecdmn0m 6851*	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)
Theorem	ereldm 6852	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 6853	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 6854	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 6855	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	ecidsn 6856	An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
⊢ [𝐴] I = {𝐴}
Theorem	qseq1 6857	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶))
Theorem	qseq2 6858	Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵))
Theorem	elqsg 6859*	Closed form of elqs 6860. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
Theorem	elqs 6860*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)
Theorem	elqsi 6861*	Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)
Theorem	ecelqsg 6862	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))
Theorem	ecelqsi 6863	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))
Theorem	ecopqsi 6864	"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)
Theorem	qsexg 6865	A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)
Theorem	qsex 6866	A quotient set exists. (Contributed by NM, 14-Aug-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 / 𝑅) ∈ V
Theorem	uniqs 6867	The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴))
Theorem	qsss 6868	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.)
⊢ (𝜑 → 𝑅 Er 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Theorem	uniqs2 6869	The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
⊢ (𝜑 → 𝑅 Er 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴)
Theorem	snec 6870	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)
Theorem	ecqs 6871	Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
⊢ 𝑅 ∈ V    ⇒   ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅)
Theorem	ecid 6872	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
Theorem	ecidg 6873	A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
Theorem	qsid 6874	A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 / ◡ E ) = 𝐴
Theorem	ectocld 6875*	Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ 𝑆 = (𝐵 / 𝑅)    &   ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵) → 𝜑)    ⇒   ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑆) → 𝜓)
Theorem	ectocl 6876*	Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ 𝑆 = (𝐵 / 𝑅)    &   ⊢ ([𝑥]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ 𝐵 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑆 → 𝜓)
Theorem	elqsn0m 6877*	An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)
Theorem	elqsn0 6878	A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅)
Theorem	ecelqsdm 6879	Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
⊢ ((dom 𝑅 = 𝐴 ∧ [𝐵]𝑅 ∈ (𝐴 / 𝑅)) → 𝐵 ∈ 𝐴)
Theorem	xpider 6880	A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝐴 × 𝐴) Er 𝐴
Theorem	iinerm 6881*	The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)
Theorem	riinerm 6882*	The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵)
Theorem	erinxp 6883	A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
⊢ (𝜑 → 𝑅 Er 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Theorem	ecinxp 6884	Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 = [𝐵](𝑅 ∩ (𝐴 × 𝐴)))
Theorem	qsinxp 6885	Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))))
Theorem	qsel 6886	If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
⊢ ((𝑅 Er 𝑋 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)
Theorem	qliftlem 6887*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
Theorem	qliftrel 6888*	𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ (𝜑 → 𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌))
Theorem	qliftel 6889*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴)))
Theorem	qliftel1 6890*	Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅𝐹𝐴)
Theorem	qliftfun 6891*	The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))
Theorem	qliftfund 6892*	The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	qliftfuns 6893*	The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))
Theorem	qliftf 6894*	The domain and codomain of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    ⇒   ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))
Theorem	qliftval 6895*	The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 Er 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
Theorem	ecoptocl 6896*	Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅)    &   ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑆 → 𝜓)
Theorem	2ecoptocl 6897*	Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
⊢ 𝑆 = ((𝐶 × 𝐷) / 𝑅)    &   ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝜒)
Theorem	3ecoptocl 6898*	Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
⊢ 𝑆 = ((𝐷 × 𝐷) / 𝑅)    &   ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ([⟨𝑧, 𝑤⟩]𝑅 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ ([⟨𝑣, 𝑢⟩]𝑅 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐷)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝜃)
Theorem	brecop 6899*	Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
⊢ ∼ ∈ V    &   ⊢ ∼ Er (𝐺 × 𝐺)    &   ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )    &   ⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ∼ ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ∼ ) ∧ 𝜑))}    &   ⊢ ((((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺)) ∧ ((𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([⟨𝑧, 𝑤⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ ∧ [⟨𝑣, 𝑢⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (𝜑 ↔ 𝜓)))    ⇒   ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨𝐴, 𝐵⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ 𝜓))
Theorem	eroveu 6900*	Lemma for eroprf 6902. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ 𝐽 = (𝐴 / 𝑅)    &   ⊢ 𝐾 = (𝐵 / 𝑆)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑅 Er 𝑈)    &   ⊢ (𝜑 → 𝑆 Er 𝑉)    &   ⊢ (𝜑 → 𝑇 Er 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑈)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑊)    &   ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)    &   ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑋 = [𝑝]𝑅 ∧ 𝑌 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))