Theorem List for Intuitionistic Logic Explorer - 6401-6500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | fnsnsplitdc 6401* |
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.)
|
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉})) |
|
Theorem | funresdfunsndc 6402* |
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 𝐹∀𝑦 ∈ dom 𝐹DECID 𝑥 = 𝑦 ∧ Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) = 𝐹) |
|
Theorem | nndifsnid 6403 |
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 3666
from subset to equality but the proof relies on equality being
decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
|
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
|
Theorem | nnaordi 6404 |
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))) |
|
Theorem | nnaord 6405 |
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))) |
|
Theorem | nnaordr 6406 |
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))) |
|
Theorem | nnaword 6407 |
Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.)
(Revised by Mario Carneiro, 15-Nov-2014.)
|
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) |
|
Theorem | nnacan 6408 |
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)) |
|
Theorem | nnaword1 6409 |
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)) |
|
Theorem | nnaword2 6410 |
Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.)
|
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
|
Theorem | nnawordi 6411 |
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) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))) |
|
Theorem | nnmordi 6412 |
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))) |
|
Theorem | nnmord 6413 |
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))) |
|
Theorem | nnmword 6414 |
Weak ordering property of ordinal multiplication. (Contributed by Mario
Carneiro, 17-Nov-2014.)
|
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) |
|
Theorem | nnmcan 6415 |
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)) |
|
Theorem | 1onn 6416 |
One is a natural number. (Contributed by NM, 29-Oct-1995.)
|
⊢ 1o ∈
ω |
|
Theorem | 2onn 6417 |
The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.)
|
⊢ 2o ∈
ω |
|
Theorem | 3onn 6418 |
The ordinal 3 is a natural number. (Contributed by Mario Carneiro,
5-Jan-2016.)
|
⊢ 3o ∈
ω |
|
Theorem | 4onn 6419 |
The ordinal 4 is a natural number. (Contributed by Mario Carneiro,
5-Jan-2016.)
|
⊢ 4o ∈
ω |
|
Theorem | nnm1 6420 |
Multiply an element of ω by 1o. (Contributed by Mario
Carneiro, 17-Nov-2014.)
|
⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) =
𝐴) |
|
Theorem | nnm2 6421 |
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)) |
|
Theorem | nn2m 6422 |
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)) |
|
Theorem | nnaordex 6423* |
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 6424* |
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 6425 |
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
|
|
Syntax | wer 6426 |
Extend the definition of a wff to include the equivalence predicate.
|
wff 𝑅 Er 𝐴 |
|
Syntax | cec 6427 |
Extend the definition of a class to include equivalence class.
|
class [𝐴]𝑅 |
|
Syntax | cqs 6428 |
Extend the definition of a class to include quotient set.
|
class (𝐴 / 𝑅) |
|
Definition | df-er 6429 |
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 6430 we derive a
more typical definition. We show that an equivalence relation is
reflexive, symmetric, and transitive in erref 6449, ersymb 6443, and ertr 6444.
(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 6430* |
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 6431 |
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 6430). 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 6432. (Contributed by
NM, 23-Jul-1995.)
|
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) |
|
Theorem | dfec2 6432* |
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 6433 |
An equivalence class modulo a set is a set. (Contributed by NM,
24-Jul-1995.)
|
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) |
|
Theorem | ecexr 6434 |
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 6435* |
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 6436 |
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 6437 |
Equality theorem for equivalence predicate. (Contributed by Mario
Carneiro, 12-Aug-2015.)
|
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵)) |
|
Theorem | errel 6438 |
An equivalence relation is a relation. (Contributed by Mario Carneiro,
12-Aug-2015.)
|
⊢ (𝑅 Er 𝐴 → Rel 𝑅) |
|
Theorem | erdm 6439 |
The domain of an equivalence relation. (Contributed by Mario Carneiro,
12-Aug-2015.)
|
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) |
|
Theorem | ercl 6440 |
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 6441 |
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 6442 |
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 6443 |
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 6444 |
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 6445 |
A transitivity relation for equivalences. (Contributed by Mario
Carneiro, 9-Jul-2014.)
|
⊢ (𝜑 → 𝑅 Er 𝑋)
& ⊢ (𝜑 → 𝐴𝑅𝐵)
& ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) |
|
Theorem | ertr2d 6446 |
A transitivity relation for equivalences. (Contributed by Mario
Carneiro, 9-Jul-2014.)
|
⊢ (𝜑 → 𝑅 Er 𝑋)
& ⊢ (𝜑 → 𝐴𝑅𝐵)
& ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐶𝑅𝐴) |
|
Theorem | ertr3d 6447 |
A transitivity relation for equivalences. (Contributed by Mario
Carneiro, 9-Jul-2014.)
|
⊢ (𝜑 → 𝑅 Er 𝑋)
& ⊢ (𝜑 → 𝐵𝑅𝐴)
& ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) |
|
Theorem | ertr4d 6448 |
A transitivity relation for equivalences. (Contributed by Mario
Carneiro, 9-Jul-2014.)
|
⊢ (𝜑 → 𝑅 Er 𝑋)
& ⊢ (𝜑 → 𝐴𝑅𝐵)
& ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) |
|
Theorem | erref 6449 |
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 6450 |
The converse of an equivalence relation is itself. (Contributed by
Mario Carneiro, 12-Aug-2015.)
|
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
|
Theorem | errn 6451 |
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 6452 |
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 6453 |
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 6454 |
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 6455* |
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 6456 |
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 6457* |
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 6458* |
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 6459* |
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 6460* |
Lemma for eqer 6461. (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 6461* |
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 6462 |
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 6463 |
The empty set is an equivalence relation on the empty set. (Contributed
by Mario Carneiro, 5-Sep-2015.)
|
⊢ ∅ Er ∅ |
|
Theorem | eceq1 6464 |
Equality theorem for equivalence class. (Contributed by NM,
23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) |
|
Theorem | eceq1d 6465 |
Equality theorem for equivalence class (deduction form). (Contributed
by Jim Kingdon, 31-Dec-2019.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
|
Theorem | eceq2 6466 |
Equality theorem for equivalence class. (Contributed by NM,
23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
|
Theorem | elecg 6467 |
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 6468 |
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 6469 |
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 6470 |
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 6471* |
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 6472 |
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 6473 |
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 6474 |
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 6475 |
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 6476 |
An equivalence class modulo the identity relation is a singleton.
(Contributed by NM, 24-Oct-2004.)
|
⊢ [𝐴] I = {𝐴} |
|
Theorem | qseq1 6477 |
Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) |
|
Theorem | qseq2 6478 |
Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) |
|
Theorem | elqsg 6479* |
Closed form of elqs 6480. (Contributed by Rodolfo Medina,
12-Oct-2010.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) |
|
Theorem | elqs 6480* |
Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
|
Theorem | elqsi 6481* |
Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
|
⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
|
Theorem | ecelqsg 6482 |
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 6483 |
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 6484 |
"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 6485 |
A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by
Mario Carneiro, 9-Jul-2014.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) |
|
Theorem | qsex 6486 |
A quotient set exists. (Contributed by NM, 14-Aug-1995.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 / 𝑅) ∈ V |
|
Theorem | uniqs 6487 |
The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
|
⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
|
Theorem | qsss 6488 |
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 6489 |
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 6490 |
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 6491 |
Equivalence class in terms of quotient set. (Contributed by NM,
29-Jan-1999.)
|
⊢ 𝑅 ∈ V ⇒ ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅) |
|
Theorem | ecid 6492 |
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 6493 |
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 6494 |
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 6495* |
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 6496* |
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 6497* |
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 6498 |
A quotient set doesn't contain the empty set. (Contributed by NM,
24-Aug-1995.)
|
⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → 𝐵 ≠ ∅) |
|
Theorem | ecelqsdm 6499 |
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 6500 |
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 |