Need help understanding proof of theorem

(The Principle of Mathematical Induction) For each positive integer , let be a statement. If

(1) is true and

(2) the implication

If , then

is true for ever positive integer , then is true for every positive integer .

*Proof:*

Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers for which is a false statement. Let

is false

Since is a nonempty subset of , it follows by the Well-Ordering Principle that contains . Since is true, .

Here I am lost:

Thus and . Therefore, and so is a true statement. By condition (2), is also true

and so . This however, contradicts our assumption that .

Question: What is ?