Satallax is a higher-orderautomated theorem prover.
Satallax won the THF divisionof CASC-23 in 2011.
Let A be an object. Let X and Y be members of ℘(A). Let a be a member of A. Assume X∈℘(Y) and a∈X. 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 A be an object. Then ℘(A) is an object.
The following background is necessary for the proof.
(.) Let A and B be objects. Then "A⊆B" is a proposition.
(.) Let A, B and x be objects. Assume A⊆B and x∈A. Then we know x∈B.
(.) Let A and B be objects. Assume B∈℘(A). Then we know B⊆A.
Since X∈℘(Y), we know X⊆Y. Using this and a∈X, we conclude a∈Y.
Test yourself on this item.