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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssopab2 5501 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) | ||
Theorem | ssopab2bw 5502* | Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5504 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 27-Dec-1996.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | eqopab2bw 5503* | Equivalence of ordered pair abstraction equality and biconditional. Version of eqopab2b 5507 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | ssopab2b 5504 | Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker ssopab2bw 5502 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} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | ssopab2i 5505 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} | ||
Theorem | ssopab2dv 5506* | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) | ||
Theorem | eqopab2b 5507 | Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker eqopab2bw 5503 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | opabn0 5508 | Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | opab0 5509 | Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | csbopab 5510* | Move substitution into a class abstraction. Version of csbopabgALT 5511 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} | ||
Theorem | csbopabgALT 5511* | Move substitution into a class abstraction. Version of csbopab 5510 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.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}) | ||
Theorem | csbmpt12 5512* | Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) | ||
Theorem | csbmpt2 5513* | Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) | ||
Theorem | iunopab 5514* | Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) Avoid ax-sep 5255, ax-nul 5262, ax-pr 5383. (Revised by SN, 11-Nov-2024.) |
⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | iunopabOLD 5515* | Obsolete version of iunopab 5514 as of 11-Dec-2024. (Contributed by Stefan O'Rear, 20-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | elopabr 5516* | Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.) |
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) | ||
Theorem | elopabran 5517* | Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.) |
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) | ||
Theorem | elopabrOLD 5518* | Obsolete version of elopabr 5516 as of 11-Dec-2024. (Contributed by AV, 16-Feb-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) | ||
Theorem | rbropapd 5519* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) | ||
Theorem | rbropap 5520* | 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.) |
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))) | ||
Theorem | 2rbropap 5521* | 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.) |
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃))) | ||
Theorem | 0nelopab 5522 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} | ||
Theorem | 0nelopabOLD 5523 | Obsolete version of 0nelopab 5522 as of 3-Oct-2024. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} | ||
Theorem | brabv 5524 | 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)) | ||
Theorem | pwin 5525 | 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 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) | ||
Theorem | pwssun 5526 | 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.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Theorem | pwun 5527 | 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.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Syntax | cid 5528 | Extend the definition of a class to include the identity relation. |
class I | ||
Definition | df-id 5529* | Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 29164). (Contributed by NM, 13-Aug-1995.) |
⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | ||
Theorem | dfid4 5530 | The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ I = (𝑥 ∈ V ↦ 𝑥) | ||
Theorem | dfid2 5531 |
Alternate definition of the identity relation. Instance of dfid3 5532 not
requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce
axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by
BJ, 5-Nov-2024.)
Use df-id 5529 instead to make the semantics of the constructor df-opab 5167 clearer. (New usage is discouraged.) |
⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥} | ||
Theorem | dfid3 5532 |
A stronger version of df-id 5529 that does not require 𝑥 and 𝑦 to
be disjoint. This is not the definition since, in order to pass our
definition soundness test, a definition has to have disjoint dummy
variables, see conventions 29130. The proof can be instructive in
showing
how disjoint variable conditions may be eliminated, a task that is not
necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by
Mario Carneiro, 18-Nov-2016.)
Use df-id 5529 instead to make the semantics of the constructor df-opab 5167 clearer (in usages, 𝑥, 𝑦 will typically be dummy variables, so can be assumed disjoint). (New usage is discouraged.) |
⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | ||
Theorem | dfid2OLD 5533 | Obsolete version of dfid2 5531 as of 4-Nov-2024. (Contributed by NM, 15-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥} | ||
Syntax | cep 5534 | Extend class notation to include the membership relation. |
class E | ||
Definition | df-eprel 5535* | 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 𝐵 ↔ 𝐴 ∈ 𝐵), when 𝐵 is a set (see epelg 5536). Thus, ⊢ 5 E {1, 5} (ex-eprel 29163). (Contributed by NM, 13-Aug-1995.) |
⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 5536 | The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5538 and closed form of epeli 5537. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epeli 5537 | The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5536. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | epel 5538 | 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 𝑥 ↔ 𝐴 ∈ 𝑥) | ||
Theorem | 0sn0ep 5539 | An example for the membership relation. (Contributed by AV, 19-Jun-2022.) |
⊢ ∅ E {∅} | ||
Theorem | epn0 5540 | The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.) |
⊢ E ≠ ∅ | ||
We have not yet defined relations (df-rel 5638), 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 typically it represents what we will later call a "relation". | ||
Syntax | wpo 5541 | Extend wff notation to include the strict partial ordering predicate. Read: "𝑅 is a partial order on 𝐴". |
wff 𝑅 Po 𝐴 | ||
Syntax | wor 5542 | Extend wff notation to include the strict total ordering predicate. Read: "𝑅 orders 𝐴". |
wff 𝑅 Or 𝐴 | ||
Definition | df-po 5543* | 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 29165). (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | ||
Definition | df-so 5544* | 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 11169). Equivalent to Definition 6.19(1) of [TakeutiZaring] p. 29. (Contributed by NM, 21-Jan-1996.) |
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
Theorem | poss 5545 | Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Po 𝐵 → 𝑅 Po 𝐴)) | ||
Theorem | poeq1 5546 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | ||
Theorem | poeq2 5547 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) | ||
Theorem | nfpo 5548 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | ||
Theorem | nfso 5549 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | ||
Theorem | pocl 5550 | Characteristic properties of a partial order in class notation. (Contributed by NM, 27-Mar-1997.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))) | ||
Theorem | poclOLD 5551 | Obsolete version of pocl 5550 as of 3-Oct-2024. (Contributed by NM, 27-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))) | ||
Theorem | ispod 5552* | Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ⇒ ⊢ (𝜑 → 𝑅 Po 𝐴) | ||
Theorem | swopolem 5553* | Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))) | ||
Theorem | swopo 5554* | A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → 𝑅 Po 𝐴) | ||
Theorem | poirr 5555 | A partial order is irreflexive. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | potr 5556 | A partial order is a transitive relation. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | po2nr 5557 | A partial order has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | po3nr 5558 | A partial order has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | po2ne 5559 | Two sets related by a partial order are not equal. (Contributed by AV, 13-Mar-2023.) |
⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵) | ||
Theorem | po0 5560 | Any relation is a partial order on the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ 𝑅 Po ∅ | ||
Theorem | pofun 5561* | The inverse image of a partial order is a partial order. (Contributed by Jeff Madsen, 18-Jun-2011.) |
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌} & ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌) ⇒ ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴) | ||
Theorem | sopo 5562 | A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.) |
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | ||
Theorem | soss 5563 | Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴)) | ||
Theorem | soeq1 5564 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) | ||
Theorem | soeq2 5565 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | ||
Theorem | sonr 5566 | A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | sotr 5567 | A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | solin 5568 | A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)) | ||
Theorem | so2nr 5569 | A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | so3nr 5570 | A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | sotric 5571 | A strict order relation satisfies strict trichotomy. (Contributed by NM, 19-Feb-1996.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | sotrieq 5572 | Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | sotrieq2 5573 | Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))) | ||
Theorem | soasym 5574 | Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) | ||
Theorem | sotr2 5575 | A transitivity relation. (Read 𝐵 ≤ 𝐶 and 𝐶 < 𝐷 implies 𝐵 < 𝐷.) (Contributed by Mario Carneiro, 10-May-2013.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((¬ 𝐶𝑅𝐵 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | issod 5576* | An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Po 𝐴) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ⇒ ⊢ (𝜑 → 𝑅 Or 𝐴) | ||
Theorem | issoi 5577* | An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ⇒ ⊢ 𝑅 Or 𝐴 | ||
Theorem | isso2i 5578* | Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ⇒ ⊢ 𝑅 Or 𝐴 | ||
Theorem | so0 5579 | 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 ∅ | ||
Theorem | somo 5580* | 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 → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | ||
Theorem | sotrine 5581 | Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ≠ 𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | sotr3 5582 | Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍)) | ||
Syntax | wfr 5583 | Extend wff notation to include the well-founded predicate. Read: "𝑅 is a well-founded relation on 𝐴". |
wff 𝑅 Fr 𝐴 | ||
Syntax | wse 5584 | Extend wff notation to include the set-like predicate. Read: "𝑅 is set-like on 𝐴". |
wff 𝑅 Se 𝐴 | ||
Syntax | wwe 5585 | Extend wff notation to include the well-ordering predicate. Read: "𝑅 well-orders 𝐴". |
wff 𝑅 We 𝐴 | ||
Definition | df-fr 5586* | Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5595 and dffr3 6048. A class is called well-founded when the membership relation E (see df-eprel 5535) 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((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) | ||
Definition | df-se 5587* | Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | ||
Definition | df-we 5588 | Define the well-ordering predicate. For an alternate definition, see dfwe2 7699. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | ||
Theorem | dffr6 5589* | Alternate definition of df-fr 5586. See dffr5 34105 for a definition without dummy variables (but note that their equivalence uses ax-sep 5255). (Contributed by BJ, 16-Nov-2024.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦) | ||
Theorem | frd 5590* | A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.) |
⊢ (𝜑 → 𝑅 Fr 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | fri 5591* | A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (inference form). (Contributed by BJ, 16-Nov-2024.) (Proof shortened by BJ, 19-Nov-2024.) |
⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | friOLD 5592* | Obsolete version of fri 5591 as of 16-Nov-2024. (Contributed by NM, 18-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | seex 5593* | 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 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V) | ||
Theorem | exse 5594 | Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | ||
Theorem | dffr2 5595* | 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.) Avoid ax-10 2138, ax-11 2155, ax-12 2172, but use ax-8 2109. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | ||
Theorem | dffr2ALT 5596* | Alternate proof of dffr2 5595, which avoids ax-8 2109 but requires ax-10 2138, ax-11 2155, ax-12 2172. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | ||
Theorem | frc 5597* | 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 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} = ∅) | ||
Theorem | frss 5598 | 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.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Fr 𝐵 → 𝑅 Fr 𝐴)) | ||
Theorem | sess1 5599 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | ||
Theorem | sess2 5600 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) |
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