Basic Binary search tree query
If the height is $0$ it is true.
Assume that all binary trees for which inorder is sorted and height smaller than $h$ are BST.
Consider a tree with root $x$, height $h$ and its inorder is sorted. In particular its left and right subtrees' inorder are sorted and have height smaller than $h$. Then, by the assumption, they are BST. Now, in the inorder of the given tree, the node $x$ is listed after all nodes of its left subtree. Since its inorder is sorted, then $x$ is not smaller than any node of its left subtree. Likewise, $x$ is not larger than any node of its right subtree. Therefore, the tree is a BST.
Hence, by induction on the height of the binary tree, all binary trees for which inorder is sorted are BST.