Proofs of Existence and Uniqueness

The sentence

There exists an x such that P(x)

is denoted by

The sentence

There exists exactly one x such that P(x)

is denoted by

There are two parts to proving a sentence of this form.

1. Existence Part. Prove that there is an x such that P(x) is true.
2. Uniqueness Part. Here you must prove that if there are two elements x and z such that P(x) and P(z) are each true, then x=z. Symbolically, .