If asked to guess, the OP's car is a 96-99 MY. The PATS system on these cars normally disables the starter. This gives a no crank symptom.
However, there is a version of PATS that needs an "OK to run" signal. The symptom is the car will start but shut off in 5 seconds. This was changed because ppl figured out that PATS could easily be defeated by jumpping the starter solenoid.
OBTW, taking power away from the PATS module will NOT work. Why? Because the PCM is expecting a signal from the PATS module in order to continue to work. Removing power to the PATS module has the same result as an attempt to steal.
Likely the OP has already attempted this (because the starter wouldn't work). Now the OP finds a new symptom where the motor shuts off after a few seconds.
The PATS system in a 96 or later MY can not be disabled by simple "wire cutting" means. It's far more effective to fix the system, reprogram the key, or disable with a hand held turner.
OBTW, some versions of PATS around the 96 MY can be reprogrammed with ZERO keys. The procedure takes 45 minutes to complete which makes it difficult to use to steal a car.
2003 GT Convertible (sold)
2000 GT coupe (fix up project)