Home Metamath Proof ExplorerTheorem List (p. 303 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26941) Hilbert Space Explorer (26942-28466) Users' Mathboxes (28467-42420)

Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj1418 30201 Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Theorembnj1417 30202* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜑𝑅 FrSe 𝐴)    &   (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))    &   (𝜃 ↔ (𝜑𝑥𝐴𝜒))    &   𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       (𝜑 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Theorembnj1421 30203* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   (𝜒 → Fun 𝑃)    &   (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))       (𝜒 → Fun 𝑄)

Theorembnj1444 30204* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))       (𝜌 → ∀𝑦𝜌)

Theorembnj1445 30205* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))    &   (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))    &   𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩       (𝜎 → ∀𝑑𝜎)

Theorembnj1446 30206* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))

Theorembnj1447 30207* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))

Theorembnj1448 30208* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))

Theorembnj1449 30209* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))       (𝜁 → ∀𝑓𝜁)

Theorembnj1442 30210* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))       (𝜂 → (𝑄𝑧) = (𝐺𝑊))

Theorembnj1450 30211* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))    &   (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))    &   𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩       (𝜁 → (𝑄𝑧) = (𝐺𝑊))

Theorembnj1423 30212* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))       (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))

Theorembnj1452 30213* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))       (𝜒𝐸𝐵)

Theorembnj1466 30214* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝑤𝑄 → ∀𝑓 𝑤𝑄)

Theorembnj1467 30215* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝑤𝑄 → ∀𝑑 𝑤𝑄)

Theorembnj1463 30216* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 ∈ V)    &   (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))    &   (𝜒𝑄 Fn 𝐸)    &   (𝜒𝐸𝐵)       (𝜒𝑄𝐶)

Theorembnj1489 30217* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝜒𝑄 ∈ V)

Theorembnj1491 30218* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))       ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

Theorembnj1312 30219* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))       (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Theorembnj1493 30220* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Theorembnj1497 30221* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       𝑔𝐶 Fun 𝑔

Theorembnj1498 30222* Technical lemma for bnj60 30223. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

20.4.5  Well-founded recursion, part 1 of 3

Theorembnj60 30223* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴𝐹 Fn 𝐴)

Theorembnj1514 30224* Technical lemma for bnj1500 30229. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))

Theorembnj1518 30225* Technical lemma for bnj1500 30229. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))    &   (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))       (𝜓 → ∀𝑑𝜓)

Theorembnj1519 30226* Technical lemma for bnj1500 30229. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1520 30227* Technical lemma for bnj1500 30229. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1501 30228* Technical lemma for bnj1500 30229. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))    &   (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))    &   (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))       (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

20.4.6  Well-founded recursion, part 2 of 3

Theorembnj1500 30229* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1525 30230* Technical lemma for bnj1522 30233. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   (𝜓 ↔ (𝜑𝐹𝐻))       (𝜓 → ∀𝑥𝜓)

Theorembnj1529 30231* Technical lemma for bnj1522 30233. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))

Theorembnj1523 30232* Technical lemma for bnj1522 30233. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   (𝜓 ↔ (𝜑𝐹𝐻))    &   (𝜒 ↔ (𝜓𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))    &   𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}    &   (𝜃 ↔ (𝜒𝑦𝐷 ∧ ∀𝑧𝐷 ¬ 𝑧𝑅𝑦))       ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

20.4.7  Well-founded recursion, part 3 of 3

Theorembnj1522 30233* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

20.5  Mathbox for Mario Carneiro

20.5.1  Predicate calculus with all distinct variables

Axiomax-7d 30234* Distinct variable version of ax-11 1971. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Axiomax-8d 30235* Distinct variable version of ax-7 1885. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Axiomax-9d1 30236 Distinct variable version of ax-6 1838, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥

Axiomax-9d2 30237* Distinct variable version of ax-6 1838, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

Axiomax-10d 30238* Distinct variable version of axc11n 2199. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-11d 30239* Distinct variable version of ax-12 1983. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

20.5.2  Miscellaneous stuff

Theoremquartfull 30240 The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑 → (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)) ≠ 0)    &   (𝜑 → -((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3) ≠ 0)       (𝜑 → ((((𝑋↑4) + (𝐴 · (𝑋↑3))) + ((𝐵 · (𝑋↑2)) + ((𝐶 · 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) − ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) + ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))) ∨ (𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) + ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) − (√‘((-(((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) − ((𝐵 − ((3 / 8) · (𝐴↑2))) / 2)) − ((((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8)) / 4) / ((√‘-((((2 · (𝐵 − ((3 / 8) · (𝐴↑2)))) + (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))) / (((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))) + (√‘((((-(2 · ((𝐵 − ((3 / 8) · (𝐴↑2)))↑3)) − (27 · (((𝐶 − ((𝐴 · 𝐵) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 · ((𝐵 − ((3 / 8) · (𝐴↑2))) · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4)))))))↑2) − (4 · ((((𝐵 − ((3 / 8) · (𝐴↑2)))↑2) + (12 · ((𝐷 − ((𝐶 · 𝐴) / 4)) + ((((𝐴↑2) · 𝐵) / 16) − ((3 / 256) · (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))))))

20.5.3  Derangements and the Subfactorial

Theoremderanglem 30241* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
(𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴1-1-onto𝐴𝜑)} ∈ Fin)

Theoremderangval 30242* Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐴 ∈ Fin → (𝐷𝐴) = (#‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))

Theoremderangf 30243* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       𝐷:Fin⟶ℕ0

Theoremderang0 30244* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐷‘∅) = 1

Theoremderangsn 30245* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       (𝐴𝑉 → (𝐷‘{𝐴}) = 0)

Theoremderangenlem 30246* One half of derangen 30247. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴𝐵𝐵 ∈ Fin) → (𝐷𝐴) ≤ (𝐷𝐵))

Theoremderangen 30247* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴𝐵𝐵 ∈ Fin) → (𝐷𝐴) = (𝐷𝐵))

Theoremsubfacval 30248* The subfactorial is defined as the number of derangements (see derangval 30242) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))

Theoremderangen2 30249* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝐴 ∈ Fin → (𝐷𝐴) = (𝑆‘(#‘𝐴)))

Theoremsubfacf 30250* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       𝑆:ℕ0⟶ℕ0

Theoremsubfaclefac 30251* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) ≤ (!‘𝑁))

Theoremsubfac0 30252* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑆‘0) = 1

Theoremsubfac1 30253* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑆‘1) = 0

Theoremsubfacp1lem1 30254* Lemma for subfacp1 30261. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})       (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (#‘𝐾) = (𝑁 − 1)))

Theoremsubfacp1lem2a 30255* Lemma for subfacp1 30261. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   (𝜑𝐺:𝐾1-1-onto𝐾)       (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹𝑀) = 1))

Theoremsubfacp1lem2b 30256* Lemma for subfacp1 30261. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐹 = (𝐺 ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   (𝜑𝐺:𝐾1-1-onto𝐾)       ((𝜑𝑋𝐾) → (𝐹𝑋) = (𝐺𝑋))

Theoremsubfacp1lem3 30257* Lemma for subfacp1 30261. In subfacp1lem6 30260 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) = 1)}    &   𝐶 = {𝑓 ∣ (𝑓:𝐾1-1-onto𝐾 ∧ ∀𝑦𝐾 (𝑓𝑦) ≠ 𝑦)}       (𝜑 → (#‘𝐵) = (𝑆‘(𝑁 − 1)))

Theoremsubfacp1lem4 30258* Lemma for subfacp1 30261. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) ≠ 1)}    &   𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})       (𝜑𝐹 = 𝐹)

Theoremsubfacp1lem5 30259* Lemma for subfacp1 30261. In subfacp1lem6 30260 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ (2...(𝑁 + 1)))    &   𝑀 ∈ V    &   𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})    &   𝐵 = {𝑔𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔𝑀) ≠ 1)}    &   𝐹 = (( I ↾ 𝐾) ∪ {⟨1, 𝑀⟩, ⟨𝑀, 1⟩})    &   𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}       (𝜑 → (#‘𝐵) = (𝑆𝑁))

Theoremsubfacp1lem6 30260* Lemma for subfacp1 30261. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 30257), while the subset with (𝑓𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 30259). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))    &   𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓𝑦) ≠ 𝑦)}       (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))

Theoremsubfacp1 30261* A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 30252, subfac1 30253. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆𝑁) + (𝑆‘(𝑁 − 1)))))

Theoremsubfacval2 30262* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ0 → (𝑆𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘))))

Theoremsubfaclim 30263* The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (abs‘(((!‘𝑁) / e) − (𝑆𝑁))) < (1 / 𝑁))

Theoremsubfacval3 30264* Another closed form expression for the subfactorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))       (𝑁 ∈ ℕ → (𝑆𝑁) = (⌊‘(((!‘𝑁) / e) + (1 / 2))))

Theoremderangfmla 30265* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))       ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (𝐷𝐴) = (⌊‘(((!‘(#‘𝐴)) / e) + (1 / 2))))

20.5.4  The Erdős-Szekeres theorem

Theoremerdszelem1 30266* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}       (𝑋𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹𝑋) Isom < , 𝑂 (𝑋, (𝐹𝑋)) ∧ 𝐴𝑋))

Theoremerdszelem2 30267* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}       ((# “ 𝑆) ∈ Fin ∧ (# “ 𝑆) ⊆ ℕ)

Theoremerdszelem3 30268* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))       (𝐴 ∈ (1...𝑁) → (𝐾𝐴) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ))

Theoremerdszelem4 30269* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ       ((𝜑𝐴 ∈ (1...𝑁)) → {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)})

Theoremerdszelem5 30270* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ       ((𝜑𝐴 ∈ (1...𝑁)) → (𝐾𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}))

Theoremerdszelem6 30271* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ       (𝜑𝐾:(1...𝑁)⟶ℕ)

Theoremerdszelem7 30272* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ    &   (𝜑𝐴 ∈ (1...𝑁))    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑 → ¬ (𝐾𝐴) ∈ (1...(𝑅 − 1)))       (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , 𝑂 (𝑠, (𝐹𝑠))))

Theoremerdszelem8 30273* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑂 Or ℝ    &   (𝜑𝐴 ∈ (1...𝑁))    &   (𝜑𝐵 ∈ (1...𝑁))    &   (𝜑𝐴 < 𝐵)       (𝜑 → ((𝐾𝐴) = (𝐾𝐵) → ¬ (𝐹𝐴)𝑂(𝐹𝐵)))

Theoremerdszelem9 30274* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)       (𝜑𝑇:(1...𝑁)–1-1→(ℕ × ℕ))

Theoremerdszelem10 30275* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)       (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽𝑚) ∈ (1...(𝑆 − 1))))

Theoremerdszelem11 30276* Lemma for erdsze 30277. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , < (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))    &   𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼𝑛), (𝐽𝑛)⟩)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)       (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))

Theoremerdsze 30277* The Erdős-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 − 1) · (𝑆 − 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , < order isomorphism, recalling that < is the greater-than relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹:(1...𝑁)–1-1→ℝ)    &   (𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)       (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))

Theoremerdsze2lem1 30278* Lemma for erdsze2 30280. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝐹:𝐴1-1→ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   𝑁 = ((𝑅 − 1) · (𝑆 − 1))    &   (𝜑𝑁 < (#‘𝐴))       (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1𝐴𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))

Theoremerdsze2lem2 30279* Lemma for erdsze2 30280. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝐹:𝐴1-1→ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   𝑁 = ((𝑅 − 1) · (𝑆 − 1))    &   (𝜑𝑁 < (#‘𝐴))    &   (𝜑𝐺:(1...(𝑁 + 1))–1-1𝐴)    &   (𝜑𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))       (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))

Theoremerdsze2 30280* Generalize the statement of the Erdős-Szekeres theorem erdsze 30277 to "sequences" indexed by an arbitrary subset of , which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
(𝜑𝑅 ∈ ℕ)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝐹:𝐴1-1→ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (#‘𝐴))       (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠))) ∨ (𝑆 ≤ (#‘𝑠) ∧ (𝐹𝑠) Isom < , < (𝑠, (𝐹𝑠)))))

20.5.5  The Kuratowski closure-complement theorem

Theoremkur14lem1 30281 Lemma for kur14 30291. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐴𝑋    &   (𝑋𝐴) ∈ 𝑇    &   (𝐾𝐴) ∈ 𝑇       (𝑁 = 𝐴 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))

Theoremkur14lem2 30282 Lemma for kur14 30291. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋       (𝐼𝐴) = (𝑋 ∖ (𝐾‘(𝑋𝐴)))

Theoremkur14lem3 30283 Lemma for kur14 30291. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋       (𝐾𝐴) ⊆ 𝑋

Theoremkur14lem4 30284 Lemma for kur14 30291. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋       (𝑋 ∖ (𝑋𝐴)) = 𝐴

Theoremkur14lem5 30285 Lemma for kur14 30291. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋       (𝐾‘(𝐾𝐴)) = (𝐾𝐴)

Theoremkur14lem6 30286 Lemma for kur14 30291. If 𝑘 is the complementation operator and 𝑘 is the closure operator, this expresses the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴 for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of 𝑐𝑘𝑐𝑘... that arise when applying closure and complement repeatedly to 𝐴, and explains why we end up with a number as large as 14, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋    &   𝐵 = (𝑋 ∖ (𝐾𝐴))       (𝐾‘(𝐼‘(𝐾𝐵))) = (𝐾𝐵)

Theoremkur14lem7 30287 Lemma for kur14 30291: main proof. The set 𝑇 here contains all the distinct combinations of 𝑘 and 𝑐 that can arise, and we prove here that applying 𝑘 or 𝑐 to any element of 𝑇 yields another elemnt of 𝑇. In operator shorthand, we have 𝑇 = {𝐴, 𝑐𝐴, 𝑘𝐴 , 𝑐𝑘𝐴, 𝑘𝑐𝐴, 𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝑐𝐴}. From the identities 𝑐𝑐𝐴 = 𝐴 and 𝑘𝑘𝐴 = 𝑘𝐴, we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴, proved in kur14lem6 30286. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋    &   𝐵 = (𝑋 ∖ (𝐾𝐴))    &   𝐶 = (𝐾‘(𝑋𝐴))    &   𝐷 = (𝐼‘(𝐾𝐴))    &   𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))       (𝑁𝑇 → (𝑁𝑋 ∧ {(𝑋𝑁), (𝐾𝑁)} ⊆ 𝑇))

Theoremkur14lem8 30288 Lemma for kur14 30291. Show that the set 𝑇 contains at most 14 elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of 14 is tight in the sense that there exist topological spaces and subsets of these spaces for which all 14 generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋    &   𝐵 = (𝑋 ∖ (𝐾𝐴))    &   𝐶 = (𝐾‘(𝑋𝐴))    &   𝐷 = (𝐼‘(𝐾𝐴))    &   𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))       (𝑇 ∈ Fin ∧ (#‘𝑇) ≤ 14)

Theoremkur14lem9 30289* Lemma for kur14 30291. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝐴𝑋    &   𝐵 = (𝑋 ∖ (𝐾𝐴))    &   𝐶 = (𝐾‘(𝑋𝐴))    &   𝐷 = (𝐼‘(𝐾𝐴))    &   𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))    &   𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}       (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14)

Theoremkur14lem10 30290* Lemma for kur14 30291. Discharge the set 𝑇. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}    &   𝐴𝑋       (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14)

Theoremkur14 30291* Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14))

20.5.6  Retracts and sections

Syntaxcretr 30292 Extend class notation with the retract relation.
class Retr

Definitiondf-retr 30293* Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽𝐾, 𝑆:𝐾𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟𝑠)(𝑗 Htpy 𝑗)( I ↾ 𝑗)) ≠ ∅})

20.5.7  Path-connected and simply connected spaces

Syntaxcpcon 30294 Extend class notation with the class of path-connected topologies.
class PCon

Syntaxcscon 30295 Extend class notation with the class of simply connected topologies.
class SCon

Definitiondf-pcon 30296* Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}

Definitiondf-scon 30297* Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.)
SCon = {𝑗 ∈ PCon ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}

Theoremispcon 30298* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽       (𝐽 ∈ PCon ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))

Theorempconcn 30299* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽       ((𝐽 ∈ PCon ∧ 𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))

Theorempcontop 30300 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ PCon → 𝐽 ∈ Top)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42420
 Copyright terms: Public domain < Previous  Next >