You need a ticket number to travel; most tickets today are "e-tickets" and not the old paper tickets.
An airline ticket is a document (can be a "virtual" document such as an e-ticket), issued by an airline or a travel agency, to confirm that an individual has purchased a seat on a flight on an aircraft. This document is then used to obtain a boarding pass, at the airport. Then with the boarding pass and the attached ticket, the passenger is allowed to board the aircraft.
Ticket numbers are comprised of thirteen digits (the first three are the ticketing or "plating" airline's airline ticketing code, beginning with American Airlines' "001") and are discrete numbers, never reused and tied to specific tickets (with their fare conditions).
Tickets issued by Aer Lingus begin with the airline ticket code "053" followed by ten digits.
Tickets issued by Virgin Atlantic begin with "932", followed by ten digits.
If your flights are
separately ticketed, you'll have no problems dropping the final segment.
(I think you're confusing a ticket with a PNR, or Passenger Name Document, usually six or seven characters (e.g. FV4UX2 or UVXTVR) that form in essence a "handle" for an itinerary. A PNR is not evidence of ticketing, and PNRs are not permanent records - they in fact are eventually "recycled" and attached to another itinerary.)
If your flights are all on one
ticket, it may not be so easy, as others have mentioned.