Automated Reasoning in Higher Order Logic
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.
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.
We have ∅∈∅ implies false. Using this, we conclude ¬∅∈∅.
Test yourself on this item.