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Theorem List for Metamath Proof Explorer - 32101-32200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj900 32101* Technical lemma for bnj69 32180. 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.)
𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝑓𝐵 → ∅ ∈ dom 𝑓)
 
Theorembnj906 32102 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj908 32103* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃!𝑓(𝑓 Fn 𝑛𝜑𝜓)))    &   (𝜃 ↔ ∀𝑚𝐷 (𝑚 E 𝑛[𝑚 / 𝑛]𝜒))    &   (𝜑′[𝑚 / 𝑛]𝜑)    &   (𝜓′[𝑚 / 𝑛]𝜓)    &   (𝜒′[𝑚 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑)    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   (𝜒″[𝐺 / 𝑓]𝜒)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))    &   (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 = suc 𝑖))    &   (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖𝑛𝑚 ≠ suc 𝑖))    &   𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐶 = 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)    &   𝐿 = 𝑦 ∈ (𝐺𝑝) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})       ((𝑅 FrSe 𝐴𝑥𝐴𝜒′𝜂) → ∃𝑓(𝐺 Fn 𝑛𝜑″𝜓″))
 
Theorembnj911 32104* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
 
Theorembnj916 32105* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))       (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
 
Theorembnj917 32106* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
 
Theorembnj934 32107* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐺 ∈ V       ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
 
Theorembnj929 32108* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐷 = (ω ∖ {∅})    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   𝐶 ∈ V       ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″)
 
Theorembnj938 32109* Technical lemma for bnj69 32180. 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.)
𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))    &   (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))    &   (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
 
Theorembnj944 32110* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜑″)
 
Theorembnj953 32111 Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))       (((𝐺𝑖) = (𝑓𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
 
Theorembnj958 32112* Technical lemma for bnj69 32180. 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.)
𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝐺𝑖) = (𝑓𝑖) → ∀𝑦(𝐺𝑖) = (𝑓𝑖))
 
Theorembnj1000 32113* Technical lemma for bnj852 32093. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐺 ∈ V    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
 
Theorembnj965 32114* Technical lemma for bnj852 32093. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓″[𝐺 / 𝑓]𝜓)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
 
Theorembnj964 32115* Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
 
Theorembnj966 32116* Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
 
Theorembnj967 32117* Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
 
Theorembnj969 32118* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
 
Theorembnj970 32119 Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝑝𝐷)
 
Theorembnj910 32120* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))    &   (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))       (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜒″)
 
Theorembnj978 32121* Technical lemma for bnj69 32180. 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.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
 
Theorembnj981 32122* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
 
Theorembnj983 32123* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
 
Theorembnj984 32124 Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝐺𝐴 → (𝐺𝐵[𝐺 / 𝑓]𝑛𝜒))
 
Theorembnj985 32125* Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       (𝐺𝐵 ↔ ∃𝑝𝜒″)
 
Theorembnj986 32126* Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))       (𝜒 → ∃𝑚𝑝𝜏)
 
Theorembnj996 32127* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
 
Theorembnj998 32128* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝜃𝜒𝜏𝜂) → 𝜒″)
 
Theorembnj999 32129* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝𝑦 ∈ (𝐺𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
 
Theorembnj1001 32130 Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   ((𝜃𝜒𝜏𝜂) → 𝜒″)       ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
 
Theorembnj1006 32131* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))       ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
 
Theorembnj1014 32132* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       ((𝑔𝐵𝑗 ∈ dom 𝑔) → (𝑔𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj1015 32133* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐺𝑉    &   𝐽𝑉       ((𝐺𝐵𝐽 ∈ dom 𝐺) → (𝐺𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj1018 32134* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))    &   ((𝜃𝜒𝜏𝜂) → 𝜒″)    &   ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))       ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj1020 32135* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))       ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj1021 32136* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
 
Theorembnj907 32137* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))    &   (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))    &   (𝜑′[𝑝 / 𝑛]𝜑)    &   (𝜓′[𝑝 / 𝑛]𝜓)    &   (𝜒′[𝑝 / 𝑛]𝜒)    &   (𝜑″[𝐺 / 𝑓]𝜑′)    &   (𝜓″[𝐺 / 𝑓]𝜓′)    &   (𝜒″[𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
 
Theorembnj1029 32138 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
 
Theorembnj1033 32139* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜂𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1034 32140* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1039 32141 Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜓′[𝑗 / 𝑖]𝜓)       (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
 
Theorembnj1040 32142* Technical lemma for bnj69 32180. 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.)
(𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜒′[𝑗 / 𝑖]𝜒)       (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
 
Theorembnj1047 32143 Technical lemma for bnj69 32180. 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.)
(𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜂′[𝑗 / 𝑖]𝜂)       (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖𝜂′))
 
Theorembnj1049 32144 Technical lemma for bnj69 32180. 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.)
(𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))       (∀𝑖𝑛 𝜂 ↔ ∀𝑖𝜂)
 
Theorembnj1052 32145* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   ((𝜃𝜏𝜒𝜁) → ( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1053 32146* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝜃𝜏𝜒𝜁) → 𝑧𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1071 32147 Technical lemma for bnj69 32180. 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.)
𝐷 = (ω ∖ {∅})       (𝑛𝐷 → E Fr 𝑛)
 
Theorembnj1083 32148 Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       (𝑓𝐾 ↔ ∃𝑛𝜒)
 
Theorembnj1090 32149* Technical lemma for bnj69 32180. 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.)
(𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜂′[𝑗 / 𝑖]𝜂)    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))       ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
 
Theorembnj1093 32150* Technical lemma for bnj69 32180. 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.)
𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
 
Theorembnj1097 32151 Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
 
Theorembnj1110 32152* Technical lemma for bnj69 32180. 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 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))       𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))
 
Theorembnj1112 32153* Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
 
Theorembnj1118 32154* Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   𝐷 = (ω ∖ {∅})    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))       𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
 
Theorembnj1121 32155 Technical lemma for bnj69 32180. 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.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
 
Theorembnj1123 32156* Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜂′[𝑗 / 𝑖]𝜂)       (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
 
Theorembnj1030 32157* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒′[𝑗 / 𝑖]𝜒)    &   (𝜃′[𝑗 / 𝑖]𝜃)    &   (𝜏′[𝑗 / 𝑖]𝜏)    &   (𝜁′[𝑗 / 𝑖]𝜁)    &   (𝜂′[𝑗 / 𝑖]𝜂)    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1124 32158 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
 
Theorembnj1133 32159* Technical lemma for bnj69 32180. 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.)
𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))    &   ((𝑖𝑛𝜏) → 𝜃)       (𝜒 → ∀𝑖𝑛 𝜃)
 
Theorembnj1128 32160* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))    &   (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))    &   (𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒′[𝑗 / 𝑖]𝜒)    &   (𝜃′[𝑗 / 𝑖]𝜃)       (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
 
Theorembnj1127 32161 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
 
Theorembnj1125 32162 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
Theorembnj1145 32163* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))        trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
 
Theorembnj1147 32164 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
 
Theorembnj1137 32165* Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))
 
Theorembnj1148 32166 Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
 
Theorembnj1136 32167* Technical lemma for bnj69 32180. 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 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)
 
Theorembnj1152 32168 Technical lemma for bnj69 32180. 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.)
(𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
 
Theorembnj1154 32169* Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theorembnj1171 32170 Technical lemma for bnj69 32180. 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.)
((𝜑𝜓) → 𝐵𝐴)    &   𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))       𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))
 
Theorembnj1172 32171 Technical lemma for bnj69 32180. 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.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))    &   ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))       𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
 
Theorembnj1173 32172 Technical lemma for bnj69 32180. 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.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))    &   ((𝜑𝜓) → 𝑅 FrSe 𝐴)    &   ((𝜑𝜓) → 𝑋𝐴)       ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
 
Theorembnj1174 32173 Technical lemma for bnj69 32180. 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.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))    &   (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))       𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
 
Theorembnj1175 32174 Technical lemma for bnj69 32180. 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.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ (𝑤𝐴𝑤𝑅𝑧)))    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))       (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
 
Theorembnj1176 32175* Technical lemma for bnj69 32180. 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.)
((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))    &   ((𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V) → ∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧)       𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
 
Theorembnj1177 32176 Technical lemma for bnj69 32180. 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.)
(𝜓 ↔ (𝑋𝐵𝑦𝐵𝑦𝑅𝑋))    &   𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ((𝜑𝜓) → 𝑅 FrSe 𝐴)    &   ((𝜑𝜓) → 𝐵𝐴)    &   ((𝜑𝜓) → 𝑋𝐴)       ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
 
Theorembnj1186 32177* Technical lemma for bnj69 32180. 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.)
𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))       ((𝜑𝜓) → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧)
 
Theorembnj1190 32178* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))    &   (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))       ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
 
Theorembnj1189 32179* Technical lemma for bnj69 32180. 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.)
(𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))    &   (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))    &   (𝜒 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)       (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
20.4.3  The existence of a minimal element in certain classes
 
Theorembnj69 32180* Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. 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.)
((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theorembnj1228 32181* Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. 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.)
(𝑤𝐵 → ∀𝑥 𝑤𝐵)       ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
20.4.4  Well-founded induction
 
Theorembnj1204 32182* Well-founded induction. 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.)
(𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))       ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
 
Theorembnj1234 32183* Technical lemma for bnj60 32232. 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.)
𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}       𝐶 = 𝐷
 
Theorembnj1245 32184* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   𝑍 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐾 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍))}       (𝜑 → dom 𝐴)
 
Theorembnj1256 32185* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)
 
Theorembnj1259 32186* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑 → ∃𝑑𝐵 Fn 𝑑)
 
Theorembnj1253 32187* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑𝐸 ≠ ∅)
 
Theorembnj1279 32188* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)
 
Theorembnj1286 32189* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)
 
Theorembnj1280 32190* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)       (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
 
Theorembnj1296 32191* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))    &   𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}    &   𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}       (𝜓 → (𝑔𝑥) = (𝑥))
 
Theorembnj1309 32192* Technical lemma for bnj60 32232. 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)}       (𝑤𝐵 → ∀𝑥 𝑤𝐵)
 
Theorembnj1307 32193* Technical lemma for bnj60 32232. 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.)
𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝑤𝐵 → ∀𝑥 𝑤𝐵)       (𝑤𝐶 → ∀𝑥 𝑤𝐶)
 
Theorembnj1311 32194* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )       ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
 
Theorembnj1318 32195 Technical lemma for bnj60 32232. 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.)
(𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))
 
Theorembnj1326 32196* Technical lemma for bnj60 32232. 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))}    &   𝐷 = (dom 𝑔 ∩ dom )       ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))
 
Theorembnj1321 32197* Technical lemma for bnj60 32232. 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))))       ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)
 
Theorembnj1364 32198 Property of FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑅 FrSe 𝐴𝑅 Se 𝐴)
 
Theorembnj1371 32199* Technical lemma for bnj60 32232. 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    &   (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))       (𝑓𝐻 → Fun 𝑓)
 
Theorembnj1373 32200* Technical lemma for bnj60 32232. 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))))    &   (𝜏′[𝑦 / 𝑥]𝜏)       (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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