Does a divorce, documented as "the first irrevocable divorce," take effect even if the husband believes it is the third divorce? And is it permissible for the couple to reconcile after that divorce? And does the final utterance, "You are divorced with a first divorce," enact a divorce?

You have divorced your wife with three pronouncements of divorce. Your statement in the third pronouncement, "I divorce you with a first divorce," does not change its reality as a third divorce. Thus, your wife has become irrevocably divorced from you (baynunah kubra), and you cannot return to her unless she marries another man with the intention of a true marriage, and then that husband consummates the marriage and divorces her or dies, and her iddah (waiting period) from him expires. If you have any doubt about the occurrence of the two previous divorces, present your case to the Sharia court or to a trustworthy scholar.