Theorem

¬∅∈∅.

Background

The following background is necessary to understand this theorem.

(.) Let A and x be objects. Then x∈A is a proposition.

(.) Let φ be a proposition. Then ¬φ is a proposition.

(.) ∅ is an object.

Background for Proof

The following background is necessary for the proof.

(.) false is the proposition given by ∅∈∅.

(.) Let φ be a proposition. Assume φ implies false. Then we know ¬φ.

Proof

We have ∅∈∅ implies false. Using this, we conclude ¬∅∈∅.

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