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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssopab2 5001 | 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 | ssopab2b 5002 | 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} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | ssopab2i 5003 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} | ||
Theorem | ssopab2dv 5004* | 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 5005 | Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | opabn0 5006 | Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | opab0 5007 | Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | csbopab 5008* | Move substitution into a class abstraction. Version of csbopabgALT 5009 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 5009* | Move substitution into a class abstraction. Version of csbopab 5008 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 5010* | 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 5011* | 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 5012* | Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | elopabr 5013* | Membership in a class abstraction of pairs, defined by a binary relation. (Contributed by AV, 16-Feb-2021.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) | ||
Theorem | elopabran 5014* | Membership in a class abstraction of pairs, defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) | ||
Theorem | rbropapd 5015* | 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 5016* | 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 5017* | 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 | pwin 5018 | 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 | pwunss 5019 | 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 (𝐴 ∪ 𝐵) | ||
Theorem | pwssun 5020 | 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 | pwundif 5021 | 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 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | pwun 5022 | 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 5023 | Extend the definition of a class to include the identity relation. |
class I | ||
Definition | df-id 5024* | Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 27291). (Contributed by NM, 13-Aug-1995.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
Theorem | dfid3 5025 | A stronger version of df-id 5024 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} | ||
Theorem | dfid4 5026 | The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ I = (𝑥 ∈ V ↦ 𝑥) | ||
Theorem | dfid2 5027 | Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.) |
⊢ I = {〈𝑥, 𝑥〉 ∣ 𝑥 = 𝑥} | ||
Syntax | cep 5028 | Extend class notation to include the membership/epsilon relation. |
class E | ||
Definition | df-eprel 5029* | Define the membership relation, or epsilon relation. Similar to Definition 6.22 of [TakeutiZaring] p. 30. The epsilon relation and set membership are the same, that is, (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) when 𝐵 is a set by epelg 5030. Thus, 5 E {1, 5} (ex-eprel 27290). (Contributed by NM, 13-Aug-1995.) |
⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 5030 | The epsilon relation and membership are the same. General version of epel 5032. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epelc 5031 | The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | epel 5032 | The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | ||
Syntax | wpo 5033 | Extend wff notation to include the strict partial ordering predicate. Read: ' 𝑅 is a partial order on 𝐴.' |
wff 𝑅 Po 𝐴 | ||
Syntax | wor 5034 | Extend wff notation to include the strict complete ordering predicate. Read: ' 𝑅 orders 𝐴.' |
wff 𝑅 Or 𝐴 | ||
Definition | df-po 5035* | 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 27292). (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | ||
Definition | df-so 5036* | 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 10118). 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 5037 | 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 5038 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | ||
Theorem | poeq2 5039 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) | ||
Theorem | nfpo 5040 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | ||
Theorem | nfso 5041 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | ||
Theorem | pocl 5042 | Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))) | ||
Theorem | ispod 5043* | 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 5044* | 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 5045* | 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 5046 | A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | potr 5047 | A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | ||
Theorem | po2nr 5048 | A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | po3nr 5049 | A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵)) | ||
Theorem | po0 5050 | 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 ∅ | ||
Theorem | pofun 5051* | A function preserves a partial order relation. (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 5052 | A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.) |
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | ||
Theorem | soss 5053 | 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 5054 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) | ||
Theorem | soeq2 5055 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | ||
Theorem | sonr 5056 | A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | sotr 5057 | 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 5058 | 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 5059 | 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 5060 | 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 5061 | 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 5062 | 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 5063 | Trichotomy law for strict order relation. (Contributed by NM, 5-May-1999.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ (¬ 𝐵𝑅𝐶 ∧ ¬ 𝐶𝑅𝐵))) | ||
Theorem | sotr2 5064 | 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 5065* | 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 5066* | 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 5067* | 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 5068 | 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 5069* | 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) | ||
Syntax | wfr 5070 | Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.' |
wff 𝑅 Fr 𝐴 | ||
Syntax | wse 5071 | Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.' |
wff 𝑅 Se 𝐴 | ||
Syntax | wwe 5072 | Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.' |
wff 𝑅 We 𝐴 | ||
Definition | df-fr 5073* | Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5079 and dffr3 5498. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) | ||
Definition | df-se 5074* | Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) | ||
Definition | df-we 5075 | Define the well-ordering predicate. For an alternate definition, see dfwe2 6981. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | ||
Theorem | fri 5076* | Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.) |
⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | seex 5077* | 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 5078 | Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | ||
Theorem | dffr2 5079* | 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((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅)) | ||
Theorem | frc 5080* | 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 5081 | 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 5082 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | ||
Theorem | sess2 5083 | Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) | ||
Theorem | freq1 5084 | Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴)) | ||
Theorem | freq2 5085 | Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) | ||
Theorem | seeq1 5086 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) | ||
Theorem | seeq2 5087 | Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵)) | ||
Theorem | nffr 5088 | Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Fr 𝐴 | ||
Theorem | nfse 5089 | Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Se 𝐴 | ||
Theorem | nfwe 5090 | Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 We 𝐴 | ||
Theorem | frirr 5091 | A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | fr2nr 5092 | A well-founded relation has no 2-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) | ||
Theorem | fr0 5093 | Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.) |
⊢ 𝑅 Fr ∅ | ||
Theorem | frminex 5094* | If an element of a well-founded set satisfies a property 𝜑, then there is a minimal element that satisfies 𝜑. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑅 Fr 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → ¬ 𝑦𝑅𝑥)))) | ||
Theorem | efrirr 5095 | Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.) |
⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | efrn2lp 5096 | A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.) |
⊢ (( E Fr 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵)) | ||
Theorem | epse 5097 | The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.) |
⊢ E Se 𝐴 | ||
Theorem | tz7.2 5098 | Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.) |
⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | ||
Theorem | dfepfr 5099* | An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.) |
⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)) | ||
Theorem | epfrc 5100* | A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (( E Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (𝐵 ∩ 𝑥) = ∅) |
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