Found very prominent and hard vein inside breast on upper outside part feels little sore, like a hard cord what can it be?

Thrombophlebitis: This sounds like a superficial vein has developed a blood clot. You need to see your physician. For evaluation. An ultrasound can determine if the clot is superficial or is actually in deeper vein. The treatment may range from simple anti-inflammatory, local heat, compression, thrombolytic, or anticoagulation therapy.

Mondor Disease: Sounds like it might be Mondor Disease, which is superficial thrombophlebitis of a breast vein. It is a benign self limiting condition, generally resolving gradually in 4-6 weeks. Anti-inflammatory medications are sometimes given for symptomatic relief. See your doctor for an evaluation, and to rule out other causes.

See your doctor and: have it checked and you may need a mammogram or ultrasound.

Breast: Start with a mammogram and then a breast ultrasound would be helpful. What you're describing sounds like Mondor's disease (thrombosis/clotting of a breast vein). Please see your doctor.