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286: ◆Ph05QxAcng  [] 2024/03/09(土) 19:56:24.91 ID:P9QGsLiu

>>285

Theorem
Existence is a silhouette.
Proof
First, we will demonstrate two lemmas.

Lemma 1.1: Consider an element 'a' and its power set {a}. These two are different entities.
Proof
Assume they are the same.
If there exist elements a, b, then the set {a, b} would be the same entity. The latter is seen as a single entity consisting of a and b, which are clearly different. Thus, the initial premise is contradicted, proving the lemma.

Lemma 1.2: Elements of a set that are elements of space are themselves elements of space.
Proof
Let a set be E, and suppose an element e of E is not an element of space. If E∩e = e, then despite e being a part of E, there would exist a part that is not an element of space, which is a contradiction. Therefore, the lemma is proven.

Taking the contrapositive, we obtain the following corollary.

Corollary 1.2.1: A set composed of elements that are not elements of space is not an element of space.

http://rio2016.5ch.net/test/read.cgi/math/1668766352/286

287: ◆Ph05QxAcng  [] 2024/03/09(土) 19:56:37.16 ID:P9QGsLiu

>>286

Next, we demonstrate that the existence of A and A having elements in space are equivalent.

Suppose there exists an entity A that exists but does not have elements in space.
Consider A and its power set {A}.
If we assume that {A} also does not have elements in space,
and define B as a set of entities that do not have elements in space. Considering {B}, if {B} is assumed to not have elements in space, then B⊇{B} and {B}⊇B would mean B and {B} coincide. However, this contradicts Lemma 1.1.

Therefore, {B} is an element of space. Thus, by Lemma 1.2, both A and {A} are shown to be elements of space, which contradicts the premise.
Hence, {A} is an element of space.
Furthermore, from Lemma 1.2, it can be deduced that A is also an element of space.

Therefore, if A exists, it is shown that A has elements in space.

Since it is self-evident that having elements in space implies existence, it is demonstrated that existence and having elements in space are equivalent. Furthermore, having elements in space is equivalent to having a contour, thus existing is equivalent to having a silhouette.

http://rio2016.5ch.net/test/read.cgi/math/1668766352/287

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