Is the mean equal to the median if the data set consist of 1000 values that are all different?

No. Not necessarily. Here are two examples with a smaller set of numbers:

A) 1, 2, 3, 4

Mean: 10/4 = 2.5

Median: the mean of the two central elements - also 2.5.

B) 1, 2, 3, 394

Mean: 400 / 4 = 100

Median: the mean of the two central elements = 2.5.

You can extend the same principle - of having numbers on side of the center farther from the center than the numbers on the other side - to get a different mean and median, for sets of just about any size.