It is true that many casinos can lawfully operate by the provisions of tribal law, in addition to or even-- in certain circumstances-- in place of state or federal law. But the circumstances of termination that you describe here would not be unlawful by state or federal law, so there seems to be no disadvantage to you in the application of tribal law. It is virtually certain that tribal law is not more restrictive of the employer's termination powers than state and federal law.
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Some casinos have an internal appeal policy in place which would allow you to appeal your termination and present evidence on your behalf. Failing that, I often refer clients to attorneys who specialize in tribal law when there is a true wrongful termination situation. But I agree with the other attorney insofar as the situation you describe does not sound like wrongful termination even under Washington law. Washington is an at-will employment state and an employer can terminate you for any reason - even a mistaken belief you did something wrong - so long as they were not discriminatory in doing so.
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