Is an exon part of an intron?

No. Those are two different parts of pre-mRNA before the transcription is finished. The intron is the non coded region, and is therefore spliced out of the mRNA.

The coding portions of eukaryotic genes are split in to two types of DNA, introns and exons. Only the exons code for the protein itself. The introns often contain control regions and are 'spliced out' in a process known as post-transcriptional modification.

It's actually a little more complicated than that in practice, as some genes have exons which they sometimes include in the mRNA that goes for transcription and at other times they won't include those exons, they'll splice them out in the same way as they would an intron. An example of this would be the cartilage structural protein collagen II.