Proving this can be challenging, but is far from impossible with an experienced employment law attorney. To prove that the reason you were not promoted is that of an illegal reason, you must show that the protected traits were substantial motivating reasons for the decision not to promote.
Evidence to prove the causal connection could include:
Patterns of the discriminatory conduct
Comments made by supervisors or other workers, or
Documents, emails, texts, or other evidence identifying discriminatory intent towards you or members of the protected class generally.