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Not exactly. A pi bond has two electrons that lie on two lobes, one on each side of the internuclear axis, but is still only one bond. P-orbitals in a pi bond overlap in a side by side fashion. A sigma bond results from end-to end overlap (consequently because of this a sigma bond is stronger than a pi bond). There cannot be a pi bond without a sigma bond, but the sigma bond is not part of the pi bond, they are the result of separate overlapping orbitals. A double bond contains one sigma bond and one separate pi bond. A triple bond contains one sigma bond and two pi bonds, so a pi bond could not contain a sigma bond otherwise a triple bond wouldn't be possible
Double bonds consist of one pi and one sigma bond because the orientation of a double bond consists of a both a side to side overlap as well as a end-to-end overlap. When you continue to add bonds, you'll add another pi bond since there is now another side-to-side overlap.
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