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Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopelopab2 5401* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜒))

Theoremopelopab 5402* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Theorembrab 5403* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵𝜒)

Theoremopelopabaf 5404* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5402 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝑥𝜓    &   𝑦𝜓    &   𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Theoremopelopabf 5405* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5402 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
𝑥𝜓    &   𝑦𝜒    &   𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Theoremssopab2 5406 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
(∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremssopab2bw 5407* Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5409 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by NM, 27-Dec-1996.) (Revised by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremeqopab2bw 5408* Equivalence of ordered pair abstraction equality and biconditional. Version of eqopab2b 5412 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Mario Carneiro, 4-Jan-2017.) (Revised by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremssopab2b 5409 Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker ssopab2bw 5407 when possible. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremssopab2i 5410 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Theoremssopab2dv 5411* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Theoremeqopab2b 5412 Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker eqopab2bw 5408 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremopabn0 5413 Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)

Theoremopab0 5414 Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremcsbopab 5415* Move substitution into a class abstraction. Version of csbopabgALT 5416 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.)
𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}

TheoremcsbopabgALT 5416* Move substitution into a class abstraction. Version of csbopab 5415 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})

Theoremcsbmpt12 5417* Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))

Theoremcsbmpt2 5418* Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))

Theoremiunopab 5419* Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 𝜑}

Theoremelopabr 5420* Membership in a class abstraction of pairs, defined by a binary relation. (Contributed by AV, 16-Feb-2021.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)

Theoremelopabran 5421* Membership in a class abstraction of pairs, defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝜓)} → 𝐴𝑅)

Theoremrbropapd 5422* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})    &   ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))       (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))

Theoremrbropap 5423* Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.)
(𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})    &   ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))       ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒)))

Theorem2rbropap 5424* Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.)
(𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓𝜏)})    &   ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))    &   ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜏𝜃))       ((𝜑𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒𝜃)))

Theorem0nelopab 5425 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Theorembrabv 5426 If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
(𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))

2.3.5  Power class of union and intersection

Theorempwin 5427 The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)

TheorempwunssOLD 5428 Obsolete version of pwunss 4532 as of 30-Dec-2023. (Contributed by NM, 23-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)

Theorempwssun 5429 The power class of the union of two classes is a subset of the union of their power classes, iff one class is a subclass of the other. Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

TheorempwundifOLD 5430 Obsolete proof of pwundif 4538 as of 26-Dec-2023. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Theorempwun 5431 The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐵𝐴) ↔ 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵))

2.3.6  The identity relation

Syntaxcid 5432 Extend the definition of a class to include the identity relation.
class I

Definitiondf-id 5433* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 28197). (Contributed by NM, 13-Aug-1995.)
I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Theoremdfid4 5434 The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)
I = (𝑥 ∈ V ↦ 𝑥)

Theoremdfid3 5435 A stronger version of df-id 5433 that does not require 𝑥 and 𝑦 to be disjoint. This is not the "official" definition since our definition soundness check without this requirement would be much more complex. The proof can be instructive in showing how disjoint variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.) Use directly the definition df-id 5433 when sufficient, since the derivation of dfid3 5435 is nontrivial and uses auxiliary axioms ax-10 2146 to ax-13 2391. (New usage is discouraged.)
I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Theoremdfid2 5436 Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.) Use df-id 5433 when sufficient (see comment at dfid3 5435). (New usage is discouraged.)
I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}

2.3.7  The membership relation (or epsilon relation)

Syntaxcep 5437 Extend class notation to include the membership relation.
class E

Definitiondf-eprel 5438* Define the membership relation (also called "epsilon relation" since it is sometimes denoted by the lowercase Greek letter "epsilon"). Similar to Definition 6.22 of [TakeutiZaring] p. 30. The membership relation and the membership predicate agree, that is, (𝐴 E 𝐵𝐴𝐵), when 𝐵 is a set (see epelg 5439). Thus, 5 E {1, 5} (ex-eprel 28196). (Contributed by NM, 13-Aug-1995.)
E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}

Theoremepelg 5439 The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5442 and closed form of epeli 5441. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

TheoremepelgOLD 5440 Obsolete version of epelg 5439 as of 14-Jul-2023. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Theoremepeli 5441 The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5439. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 E 𝐵𝐴𝐵)

Theoremepel 5442 The membership relation and the membership predicate agree when the "containing" class is a setvar. (Contributed by NM, 13-Aug-1995.) Replace the first setvar variable with a class variable. (Revised by BJ, 13-Sep-2022.)
(𝐴 E 𝑥𝐴𝑥)

Theorem0sn0ep 5443 An example for the membership relation. (Contributed by AV, 19-Jun-2022.)
∅ E {∅}

Theoremepn0 5444 The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.)
E ≠ ∅

2.3.8  Partial and total orderings

We have not yet defined relations (df-rel 5535), but here we introduce a few related notions we will use to develop ordinals. The class variable 𝑅 is no different from other class variables, but it reminds us that normally it represents what we will later call a "relation".

Syntaxwpo 5445 Extend wff notation to include the strict partial ordering predicate. Read: "𝑅 is a partial order on 𝐴".
wff 𝑅 Po 𝐴

Syntaxwor 5446 Extend wff notation to include the strict total ordering predicate. Read: "𝑅 orders 𝐴".
wff 𝑅 Or 𝐴

Definitiondf-po 5447* Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression 𝑅 Po 𝐴 means 𝑅 is a partial order on 𝐴. For example, < Po ℝ is true, while ≤ Po ℝ is false (ex-po 28198). (Contributed by NM, 16-Mar-1997.)
(𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

Definitiondf-so 5448* Define the strict complete (linear) order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. For example, < Or ℝ is true (ltso 10698). Equivalent to Definition 6.19(1) of [TakeutiZaring] p. 29. (Contributed by NM, 21-Jan-1996.)
(𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))

Theoremposs 5449 Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
(𝐴𝐵 → (𝑅 Po 𝐵𝑅 Po 𝐴))

Theorempoeq1 5450 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))

Theorempoeq2 5451 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
(𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))

Theoremnfpo 5452 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Po 𝐴

Theoremnfso 5453 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Or 𝐴

Theorempocl 5454 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
(𝑅 Po 𝐴 → ((𝐵𝐴𝐶𝐴𝐷𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Theoremispod 5455* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))       (𝜑𝑅 Po 𝐴)

Theoremswopolem 5456* Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))       ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))

Theoremswopo 5457* A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))       (𝜑𝑅 Po 𝐴)

Theorempoirr 5458 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Theorempotr 5459 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Theorempo2nr 5460 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Theorempo3nr 5461 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))

Theorempo2ne 5462 Two classes which are in a partial order relation are not equal. (Contributed by AV, 13-Mar-2023.)
((𝑅 Po 𝑉 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝑅𝐵) → 𝐴𝐵)

Theorempo0 5463 Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑅 Po ∅

Theorempofun 5464* A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}    &   (𝑥 = 𝑦𝑋 = 𝑌)       ((𝑅 Po 𝐵 ∧ ∀𝑥𝐴 𝑋𝐵) → 𝑆 Po 𝐴)

Theoremsopo 5465 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
(𝑅 Or 𝐴𝑅 Po 𝐴)

Theoremsoss 5466 Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))

Theoremsoeq1 5467 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))

Theoremsoeq2 5468 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
(𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))

Theoremsonr 5469 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Theoremsotr 5470 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Theoremsolin 5471 A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))

Theoremso2nr 5472 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))

Theoremso3nr 5473 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))

Theoremsotric 5474 A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))

Theoremsotrieq 5475 Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Theoremsotrieq2 5476 Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))

Theoremsoasym 5477 Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Theoremsotr2 5478 A transitivity relation. (Read 𝐵𝐶 and 𝐶 < 𝐷 implies 𝐵 < 𝐷.) (Contributed by Mario Carneiro, 10-May-2013.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((¬ 𝐶𝑅𝐵𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Theoremissod 5479* An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝜑𝑅 Po 𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))       (𝜑𝑅 Or 𝐴)

Theoremissoi 5480* An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
(𝑥𝐴 → ¬ 𝑥𝑅𝑥)    &   ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))    &   ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))       𝑅 Or 𝐴

Theoremisso2i 5481* Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))    &   ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))       𝑅 Or 𝐴

Theoremso0 5482 Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑅 Or ∅

Theoremsomo 5483* A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)
(𝑅 Or 𝐴 → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)

2.3.9  Founded and well-ordering relations

Syntaxwfr 5484 Extend wff notation to include the well-founded predicate. Read: "𝑅 is a well-founded relation on 𝐴".
wff 𝑅 Fr 𝐴

Syntaxwse 5485 Extend wff notation to include the set-like predicate. Read: "𝑅 is set-like on 𝐴".
wff 𝑅 Se 𝐴

Syntaxwwe 5486 Extend wff notation to include the well-ordering predicate. Read: "𝑅 well-orders 𝐴".
wff 𝑅 We 𝐴

Definitiondf-fr 5487* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5493 and dffr3 5935. A class is called well-founded when the membership relation E (see df-eprel 5438) is well-founded on it, that is, 𝐴 is well-founded if E Fr 𝐴 (some sources request that the membership relation be well-founded on its transitive closure). (Contributed by NM, 3-Apr-1994.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))

Definitiondf-se 5488* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)

Definitiondf-we 5489 Define the well-ordering predicate. For an alternate definition, see dfwe2 7471. (Contributed by NM, 3-Apr-1994.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))

Theoremfri 5490* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
(((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theoremseex 5491* The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)

Theoremexse 5492 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝐴𝑉𝑅 Se 𝐴)

Theoremdffr2 5493* Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))

Theoremfrc 5494* Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)
𝐵 ∈ V       ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} = ∅)

Theoremfrss 5495 Subset theorem for the well-founded predicate. Exercise 1 of [TakeutiZaring] p. 31. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))

Theoremsess1 5496 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))

Theoremsess2 5497 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))

Theoremfreq1 5498 Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))

Theoremfreq2 5499 Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
(𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))

Theoremseeq1 5500 Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅 = 𝑆 → (𝑅 Se 𝐴𝑆 Se 𝐴))

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