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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | otsndisj 5401* | The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {〈𝐴, 𝐵, 𝑐〉}) | ||
Theorem | otiunsndisj 5402* | The union of singletons consisting of ordered triples which have distinct first and third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){〈𝑎, 𝐵, 𝑐〉}) | ||
Theorem | iunopeqop 5403* | Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ ∅ → (∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = 〈𝐶, 𝐷〉 → ∃𝑧 𝐴 = {𝑧})) | ||
Theorem | opabidw 5404* | Version of opabid 5405 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | opabid 5405 | 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) | ||
Theorem | elopab 5406* | Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | ||
Theorem | rexopabb 5407* | Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.) |
⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ 𝜑} & ⊢ (𝑜 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒)) | ||
Theorem | opelopabsbALT 5408* | The law of concretion in terms of substitutions. Less general than opelopabsb 5409, 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) | ||
Theorem | opelopabsb 5409* | 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) | ||
Theorem | brabsb 5410* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | ||
Theorem | opelopabt 5411* | Closed theorem form of opelopab 5421. (Contributed by NM, 19-Feb-2013.) |
⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | opelopabga 5412* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) | ||
Theorem | brabga 5413* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) | ||
Theorem | opelopab2a 5414* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓)) | ||
Theorem | opelopaba 5415* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | braba 5416* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜓) | ||
Theorem | opelopabg 5417* | 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | brabg 5418* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | opelopabgf 5419* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5417 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | opelopab2 5420* | Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜒)) | ||
Theorem | opelopab 5421* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | brab 5422* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜒) | ||
Theorem | opelopabaf 5423* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5421 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.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | opelopabf 5424* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5421 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | ssopab2 5425 | 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 5426* | Version of ssopab2b 5428 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | eqopab2bw 5427* | Version of eqopab2b 5431 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | ssopab2b 5428 | 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 5429 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} | ||
Theorem | ssopab2dv 5430* | 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 5431 | 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 5432 | Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | opab0 5433 | Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | csbopab 5434* | Move substitution into a class abstraction. Version of csbopabgALT 5435 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 5435* | Move substitution into a class abstraction. Version of csbopab 5434 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 5436* | 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 5437* | 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 5438* | 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 5439* | 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 5440* | 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 5441* | 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 5442* | 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 5443* | 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 5444 | 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} | ||
Theorem | brabv 5445 | 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 5446 | 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 5447 | 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.) (Proof shortened by BJ, 30-Dec-2023.) |
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | pwunssOLD 5448 | 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | pwssun 5449 | 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 5450 | 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.) (Proof shortened by BJ, 26-Dec-2023.) |
⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | pwundifOLD 5451 | Obsolete proof of pwundif 5450 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 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | pwun 5452 | 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 5453 | Extend the definition of a class to include the identity relation. |
class I | ||
Definition | df-id 5454* | Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 28141). (Contributed by NM, 13-Aug-1995.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
Theorem | dfid4 5455 | The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ I = (𝑥 ∈ V ↦ 𝑥) | ||
Theorem | dfid3 5456 | A stronger version of df-id 5454 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 5454 when sufficient, since the derivation of dfid3 5456 is nontrivial and uses auxiliary axioms ax-10 2136 to ax-13 2383. (New usage is discouraged.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
Theorem | dfid2 5457 | Alternate definition of the identity relation. (Contributed by NM, 15-Mar-2007.) Use df-id 5454 when sufficient (see comment at dfid3 5456). (New usage is discouraged.) |
⊢ I = {〈𝑥, 𝑥〉 ∣ 𝑥 = 𝑥} | ||
Syntax | cep 5458 | Extend class notation to include the membership relation. |
class E | ||
Definition | df-eprel 5459* | 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 5460). Thus, ⊢ 5 E {1, 5} (ex-eprel 28140). (Contributed by NM, 13-Aug-1995.) |
⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 5460 | The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5463 and closed form of epeli 5462. (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 | epelgOLD 5461 | Obsolete version of epelg 5460 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.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epeli 5462 | The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5460. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | epel 5463 | 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 5464 | An example for the membership relation. (Contributed by AV, 19-Jun-2022.) |
⊢ ∅ E {∅} | ||
Theorem | epn0 5465 | The membership relation is nonempty. (Contributed by AV, 19-Jun-2022.) |
⊢ E ≠ ∅ | ||
We have not yet defined relations (df-rel 5556), 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". | ||
Syntax | wpo 5466 | Extend wff notation to include the strict partial ordering predicate. Read: "𝑅 is a partial order on 𝐴". |
wff 𝑅 Po 𝐴 | ||
Syntax | wor 5467 | Extend wff notation to include the strict total ordering predicate. Read: "𝑅 orders 𝐴". |
wff 𝑅 Or 𝐴 | ||
Definition | df-po 5468* | 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 28142). (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) | ||
Definition | df-so 5469* | 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 10710). 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 5470 | 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 5471 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐴)) | ||
Theorem | poeq2 5472 | Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Po 𝐴 ↔ 𝑅 Po 𝐵)) | ||
Theorem | nfpo 5473 | Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Po 𝐴 | ||
Theorem | nfso 5474 | Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝑅 Or 𝐴 | ||
Theorem | pocl 5475 | 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 5476* | 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 5477* | 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 5478* | 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 5479 | A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | potr 5480 | 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 5481 | 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 5482 | 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 | po2ne 5483 | Two classes which are in a partial order relation are not equal. (Contributed by AV, 13-Mar-2023.) |
⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵) | ||
Theorem | po0 5484 | 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 5485* | 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 5486 | A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.) |
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | ||
Theorem | soss 5487 | 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 5488 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴)) | ||
Theorem | soeq2 5489 | Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) |
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵)) | ||
Theorem | sonr 5490 | A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | ||
Theorem | sotr 5491 | 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 5492 | 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 5493 | 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 5494 | 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 5495 | 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 5496 | 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 5497 | 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 5498 | Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) | ||
Theorem | sotr2 5499 | 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 5500* | 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) |
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