ミレニアム懸賞問題

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英訳
Theorem 2: Existence is a silhouette.
Proof
First, we will demonstrate two lemmas.

Lemma 2.1: 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 2.2: 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 2.2.1: A set composed of elements that are not elements of space is not an element of space.