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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 4801 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
 
Theoremelopab 4802* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
 
TheoremopelopabsbALT 4803* The law of concretion in terms of substitutions. Less general than opelopabsb 4804, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
 
Theoremopelopabsb 4804* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
(⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
 
Theorembrabsb 4805* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
 
Theoremopelopabt 4806* Closed theorem form of opelopab 4816. (Contributed by NM, 19-Feb-2013.)
((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
 
Theoremopelopabga 4807* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
 
Theorembrabga 4808* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
 
Theoremopelopab2a 4809* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
 
Theoremopelopaba 4810* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
 
Theorembraba 4811* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵𝜓)
 
Theoremopelopabg 4812* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
 
Theorembrabg 4813* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
 
Theoremopelopabgf 4814* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4812 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
 
Theoremopelopab2 4815* 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 4816* 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 4817* 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 4818* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4816 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 4819* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4816 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 4820 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})
 
Theoremssopab2b 4821 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
 
Theoremssopab2i 4822 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
 
Theoremssopab2dv 4823* 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 4824 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
 
Theoremopabn0 4825 Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)
 
Theoremcsbopab 4826* Move substitution into a class abstraction. Version of csbopabgALT 4827 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 4827* Move substitution into a class abstraction. Version of csbopab 4826 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 4828* Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
 
Theoremcsbmpt2 4829* Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
 
Theoremiunopab 4830* 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 4831* Membership in a class abstraction of pairs, defined by a binary relation. (Contributed by AV, 16-Feb-2021.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴𝑅)
 
Theoremelopabran 4832* 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 4833* 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 4834* 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 4835* 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)))
 
2.3.5  Power class of union and intersection
 
Theorempwin 4836 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)
 
Theorempwunss 4837 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
 
Theorempwssun 4838 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))
 
Theorempwundif 4839 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)
𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 
Theorempwun 4840 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  Epsilon and identity relations
 
Syntaxcep 4841 Extend class notation to include the epsilon relation.
class E
 
Syntaxcid 4842 Extend the definition of a class to include identity relation.
class I
 
Definitiondf-eprel 4843* Define the epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is, (𝐴 E 𝐵𝐴𝐵) when 𝐵 is a set by epelg 4844. Thus, 5 E {1, 5} (ex-eprel 26420). (Contributed by NM, 13-Aug-1995.)
E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
 
Theoremepelg 4844 The epsilon relation and membership are the same. General version of epel 4846. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
 
Theoremepelc 4845 The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
𝐵 ∈ V       (𝐴 E 𝐵𝐴𝐵)
 
Theoremepel 4846 The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.)
(𝑥 E 𝑦𝑥𝑦)
 
Definitiondf-id 4847* Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 26421). (Contributed by NM, 13-Aug-1995.)
I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
 
Theoremdfid3 4848 A stronger version of df-id 4847 that doesn't require 𝑥 and 𝑦 to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct 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.)
I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
 
Theoremdfid4 4849 The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.)
I = (𝑥 ∈ V ↦ 𝑥)
 
Theoremdfid2 4850 Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.)
I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
 
2.3.7  Partial and complete ordering
 
Syntaxwpo 4851 Extend wff notation to include the strict partial ordering predicate. Read: ' 𝑅 is a partial order on 𝐴.'
wff 𝑅 Po 𝐴
 
Syntaxwor 4852 Extend wff notation to include the strict complete ordering predicate. Read: ' 𝑅 orders 𝐴.'
wff 𝑅 Or 𝐴
 
Definitiondf-po 4853* 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 26422). (Contributed by NM, 16-Mar-1997.)
(𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
 
Definitiondf-so 4854* 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 9868). 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 4855 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 4856 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
 
Theorempoeq2 4857 Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
(𝐴 = 𝐵 → (𝑅 Po 𝐴𝑅 Po 𝐵))
 
Theoremnfpo 4858 Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Po 𝐴
 
Theoremnfso 4859 Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝑥𝑅    &   𝑥𝐴       𝑥 𝑅 Or 𝐴
 
Theorempocl 4860 Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
(𝑅 Po 𝐴 → ((𝐵𝐴𝐶𝐴𝐷𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
 
Theoremispod 4861* Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
((𝜑𝑥𝐴) → ¬ 𝑥𝑅𝑥)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))       (𝜑𝑅 Po 𝐴)
 
Theoremswopolem 4862* 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 4863* 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 4864 A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 
Theorempotr 4865 A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
 
Theorempo2nr 4866 A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
 
Theorempo3nr 4867 A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
((𝑅 Po 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
 
Theorempo0 4868 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 4869* 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 4870 A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
(𝑅 Or 𝐴𝑅 Po 𝐴)
 
Theoremsoss 4871 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 4872 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
(𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
 
Theoremsoeq2 4873 Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
(𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
 
Theoremsonr 4874 A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 
Theoremsotr 4875 A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
 
Theoremsolin 4876 A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
 
Theoremso2nr 4877 A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐵))
 
Theoremso3nr 4878 A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
 
Theoremsotric 4879 A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶𝐶𝑅𝐵)))
 
Theoremsotrieq 4880 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 4881 Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵)))
 
Theoremsotr2 4882 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 4883* 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 4884* 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 4885* 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 4886 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 4887* 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)
 
2.3.8  Founded and well-ordering relations
 
Syntaxwfr 4888 Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.'
wff 𝑅 Fr 𝐴
 
Syntaxwse 4889 Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.'
wff 𝑅 Se 𝐴
 
Syntaxwwe 4890 Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.'
wff 𝑅 We 𝐴
 
Definitiondf-fr 4891* Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 4897 and dffr3 5308. (Contributed by NM, 3-Apr-1994.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
 
Definitiondf-se 4892* Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
 
Definitiondf-we 4893 Define the well-ordering predicate. For an alternate definition, see dfwe2 6749. (Contributed by NM, 3-Apr-1994.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
 
Theoremfri 4894* Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
(((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theoremseex 4895* 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 4896 Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝐴𝑉𝑅 Se 𝐴)
 
Theoremdffr2 4897* 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 4898* 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 4899 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 4900 Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
(𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
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