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Post #2566010

2026-04-27 06:39 UTC

Proposition 83, p. 65: If the image of the union of π‘ˆ and π‘Š is a subset of the union of π‘ˆ and π‘Š, 𝐴 is an element of π‘ˆ and 𝐡 follows 𝐴 in the #TransitiveClosure of 𝑅, then 𝐡 is an element of the union of π‘ˆ and π‘Š. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) ; 𝑅 is a set, i.e. an element of the universal class V (not 𝑉). Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ π‘ˆ) ; 𝐴 is an element of class π‘ˆ. Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V) ; 𝐡 is a set. Hyp. ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐡) Hyp. ⊒ (πœ‘ β†’ (𝑅 β€œ (π‘ˆ βˆͺ π‘Š)) βŠ† (π‘ˆ βˆͺ π‘Š)) ; Relation 𝑅 is hereditary in the union of classes π‘ˆ and π‘Š. Therefore ⊒ (πœ‘ β†’ 𝐡 ∈ (π‘ˆ βˆͺ π‘Š)) β€”β€”β€” Proposition 96, p. 71. If 𝐢 follows 𝐴 in the transitive closure of 𝑅 and 𝐡 follows 𝐢 in 𝑅, then 𝐡 follows 𝐴 in the transitive closure of 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐢 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐢) ; i.e. 𝐢 eventually follows 𝐴 Hyp. ⊒ (πœ‘ β†’ 𝐢𝑅𝐡) ; 𝐡 immediately follows 𝐢 Therefore ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐡) β€”β€”β€” Proposition 87, p. 66: If the images of both {𝐴} and π‘ˆ are subsets of π‘ˆ and 𝐢 follows 𝐴 in the transitive closure of 𝑅 and 𝐡 follows 𝐢 in 𝑅, then 𝐡 is an element of π‘ˆ. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V Hyp. ⊒ (πœ‘ β†’ 𝐢 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐢) Hyp. ⊒ (πœ‘ β†’ 𝐢𝑅𝐡) Hyp. ⊒ (πœ‘ β†’ (𝑅 β€œ {𝐴}) βŠ† π‘ˆ) Hyp. ⊒ (πœ‘ β†’ (𝑅 β€œ π‘ˆ) βŠ† π‘ˆ) Therefore ⊒ (πœ‘ β†’ 𝐡 ∈ π‘ˆ) β€”β€”β€” Proposition 91, p. 68. If 𝐡 follows 𝐴 in 𝑅 then 𝐡 follows 𝐴 in the transitive closure of 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴𝑅𝐡) Therefore ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐡) β€”β€”β€” Proposition 97, p. 71: If 𝐴 contains all elements after those in π‘ˆ in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 = ((tcβ€˜π‘…) β€œ π‘ˆ)) Therefore ⊒ (πœ‘ β†’ (𝑅 β€œ 𝐴) βŠ† 𝐴)

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  • @Arpie4Math@mathstodon.xyz 2026-04-27 06:41

    Proposition 98, p. 71: If 𝐢 follows 𝐴 and 𝐡 follows 𝐢 in the #TransitiveClosure of 𝑅, then 𝐡 follows 𝐴 in the transitive closure of 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐢 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐢) Hyp. ⊒ (πœ‘ β†’ 𝐢(tcβ€˜π‘…)𝐡) Therefore ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐡) β€”β€”β€” Proposition 102, p. 72: If either 𝐴 and 𝐢 are the same or 𝐢 follows 𝐴 in the transitive closure of 𝑅 and 𝐡 is the successor to 𝐢, then 𝐡 follows 𝐴 in the transitive closure of 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐢 ∈ V) Hyp. ⊒ (πœ‘ β†’ (𝐴(tcβ€˜π‘…)𝐢 ∨ 𝐴 = 𝐢)) Hyp. ⊒ (πœ‘ β†’ 𝐢𝑅𝐡) Therefore ⊒ (πœ‘ β†’ 𝐴(tcβ€˜π‘…)𝐡) β€”β€”β€” Proposition 106, p. 73: If 𝐡 follows 𝐴 in 𝑅, then either 𝐴 and 𝐡 are the same or 𝐡 follows 𝐴 in 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝐴𝑅𝐡) Therefore ⊒ (πœ‘ β†’ (𝐴𝑅𝐡 ∨ 𝐴 = 𝐡)) β€”β€”β€” Proposition 108, p. 74: If either 𝐴 and 𝐢 are the same or 𝐢 follows 𝐴 in the transitive closure of 𝑅 and 𝐡 is the successor to 𝐢, then either 𝐴 and 𝐡 are the same or 𝐡 follows 𝐴 in the transitive closure of 𝑅. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐡 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐢 ∈ V) Hyp. ⊒ (πœ‘ β†’ (𝐴(tcβ€˜π‘…)𝐢 ∨ 𝐴 = 𝐢)) Hyp. ⊒ (πœ‘ β†’ 𝐢𝑅𝐡) Therefore ⊒ (πœ‘ β†’ (𝐴(tcβ€˜π‘…)𝐡 ∨ 𝐴 = 𝐡)) β€”β€”β€” Proposition 109, p. 74: If 𝐴 contains all elements of π‘ˆ and all elements after those in π‘ˆ in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Hyp. ⊒ (πœ‘ β†’ 𝑅 ∈ V) Hyp. ⊒ (πœ‘ β†’ 𝐴 = (π‘ˆ βˆͺ ((tcβ€˜π‘…) β€œ π‘ˆ))) Therefore ⊒ (πœ‘ β†’ (𝑅 β€œ 𝐴) βŠ† 𝐴) β€”β€”β€” Proposition 114, p. 76: If either 𝑅 relates 𝐴 and 𝐡 or 𝐴 and 𝐡 are the same, then either 𝐴 and 𝐡 are the same, 𝑅 relates 𝐴 and 𝐡, 𝑅 relates 𝐡 and 𝐴. Hyp. ⊒ (πœ‘ β†’ (𝐴𝑅𝐡 ∨ 𝐴 = 𝐡)) Therefore ⊒ (πœ‘ β†’ (𝐴𝑅𝐡 ∨ 𝐴 = 𝐡 ∨ 𝐡𝑅𝐴))

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