Javascript Interactive Higher-Order Theorem Prover
Let A be an object. Let X and Y be members of ℘(A). Let a be a member of A. Assume ¬a∈X∪Y. Then we have ¬a∈Y.
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.
(.) Let A be an object. Then ℘(A) is an object.
(.) Let A and B be objects. Then "A∪B" is an object.
The following background is necessary for the proof.
(.) Let φ and ψ be propositions. Assume φ implies ψ and ¬ψ. Then we know ¬φ.
(.) Let A, B and x be objects. Assume x∈B. Then we know x∈A∪B.
We have a∈Y implies a∈X∪Y. Using this and ¬a∈X∪Y, we conclude ¬a∈Y.
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