Rephrase: Why does Elijah say he is the only prophet left, when the prophet Obadiah says he wasn't?
Quick Answer: Elijah likely said this - because he believed it to be true for the same reasons stated in the texts ...
The apparent contradiction:
NASB, 1 Kings 18:3-4 - ... (Now Obadiah feared the Lord greatly; 4 for when Jezebel destroyed the prophets of the Lord, Obadiah took a hundred prophets and hid them by fifties in a cave, and provided them with bread and water.)
NASB, 1 Kings 18:22 - Then Elijah said to the people, “I alone am left a prophet of the Lord, but Baal’s prophets are 450 men.
2. Context & Correcting Misperceptions:
Elijah wasn't in distress about the prophets of Baal:
It cannot be reasonably argued that Elijah's distress and statement about being alone had anything to do with being the "only one standing" against the prophets of Baal - far from it:
NASB, 1 Kings 18:27 - It came about at noon, that Elijah mocked them and said, “Call out with a loud voice, for he is a god; either he is occupied or gone aside, or is on a journey, or perhaps he is asleep and needs to be awakened.”
Elijah didn't actually "remain" standing - against anyone:
The text, (1 Kings 18:22, Interlinear), cannot possibly be interpreted as though Elijah believed he was the only prophet left standing against Baal, Ahab, and Jezebel. In fact, he was running for his life - just like the other "prophets" had done before.
NASB, 1 Kings 19:3 - And when he saw that, he arose and ran for his life, and went to Beersheba, which belongs to Judah, and left his servant there. ... 10. ... I alone am left; and they seek to take my life.”
So, Elijah must have been speaking about being the "only prophet" - in a different sense.
The difference between "Only" and "Alone":
At the very least, the Hebrew text supports an interpretation that Elijah might have been saying he was a prophet
alone | בַּד, (cf. "לְבַדּ֔וֹ" in 1 Kings 18:6, Interlinear), rather than an
only | בַּד prophet, (cf. "לְבַדּ֖וֹ" in 2 Samuel 13:33, Interlinear). Hermeneutically, the reader has discretion here.
NASB, 1 Kings 19:4 - But he [Elijah] .. requested for himself that he might die, and said, “It is enough; now, O Lord, take my life, for I am not better than my fathers.”
In the text, when Elijah begged God to let him die God granted his request - without correcting any of Elijah's claims.
Elijah was truly alone.
This is all consistent with the Scriptures that say say it is necessary to strengthen the weary to help prevent their own fall:
Paraphrase, Job 4:4; Hebrews 12:12 - Strengthen the hands that hang down, and the knees that are weak and feeble.
3. Elijah truly was the only - faithful - prophet left:
The inferences that Elijah was alone, and that Elijah was the only faithful prophet left, are the strongest conclusions - because they are directly supported by the text.
It is entirely plausible that Elijah was the only prophet who had not actually "bowed his knee to Baal".
In context, Elijah wasn't saying that there weren't other "prophets" in Israel, nor that there weren't other "believers" - Elijah was speaking about remaining faithful to God - never bowing having bowed his knee to Baal:
NASB, 1 Kings 19:18 - Yet I will leave 7,000 in Israel, all the knees that have not bowed to Baal and every mouth that has not kissed him.”
As a side note, statistically speaking: if it is true that God only pointed to 7000 faithful remaining - in all Israel - then it is extremely probable that Elijah truly was the only faithful prophet left.
4. Just because someone has the gifts of prophecy, doesn't make them a prophet:
According to the texts, Elijah actually was the only anointed Prophet, (a title and office of authority), at that time, regardless of whether there were others with prophetic gifts.
NASB, 1 Kings 19:16 - ... And Elisha the son of Shaphat of Abel Meholah you shall anoint as prophet in your place.
In the Old Testament, there are actually many examples of people exercising prophetic gifts but not actually being "anointed prophets", (they didn't have authority):
NASB, 1 Samuel 20:11 - ... Is Saul also among the prophets?
NASB, Joshua 13:22 - The sons of Israel also killed Balaam the son of Beor, the diviner, with the sword among the rest of their slain.
The New Testament also clearly distinguishes between the gifts of prophecy, and the office of a prophet, (in Ephesians 4, and 1 Corinthians 12-14).
In 1 Kings, the "Other Prophets" even Prophesied their Own Deaths due to Unfaithfulness:
NASB, 1 Kings 20:35 - Now a certain man of the sons of the prophets said to another by the word of the Lord, “Please strike me.” But the man refused to strike him. 36 Then he said to him, “Because you have not listened to the voice of the Lord, behold, as soon as you have departed from me, a lion will kill you.” And as soon as he had departed from him a lion found him and killed him.
5. Obadiah and the other prophets might have been dead when Elijah spoke:
The writers of the Old Testament never attempt to give complete histories, and often omit very relevant information.
Given that the entire context is about Ahab, Jezebel and rampant murder, it is entirely plausible that Obadiah was killed once he delivered Elijah's message, and then the rest of the prophets he had hid afterwards.
Either: 1.) Obadiah wasn't a "true and faithful prophet" - and Elijah was right; or 2.) Obadiah actually was a true and faithful prophet, but prophesied his own death - and Elijah was right; or 3.) Ahab and Jezebel decided to "be nice" and to let Obadiah live - believing that he had been honest with them all along, and didn't care he was a faithful prophet - and the writer of the text was incredibly inconsistent and contradictory in a very small context, (improbable).
What proves a "Prophet" as an "Anointed Prophet" - Their words never fail:
NKJV, 1 Samuel 3:19 - So Samuel grew, and the Lord was with him and let none of his words fall to the ground.
This might explain why true prophets might consider it a terrifying and shameful thing to be a prophet, (Zechariah 13:4) - and safeguard every word that comes out of their mouths.
NKJV, Deuteronomy 18:22 - when a prophet speaks in the name of the Lord, if the thing does not happen or come to pass, that is the thing which the Lord has not spoken; the prophet has spoken it presumptuously; you shall not be afraid of him [(those self-proclaimed prophets never had authority)].
Obadiah prophesied his own death:
NASB, 1 Kings 18:9 - He [Obadiah] said [to Elijah], “What sin have I committed, that you are giving your servant into the hand of Ahab to put me to death? 10 As the Lord your God lives, there is no nation or kingdom where my master has not sent to search for you; and when they said, ‘He is not here,’ he made the kingdom or nation swear that they could not find you. 11 And now you are saying, ‘Go, say to your master, “Behold, Elijah is here.”’
The prospect of inadvertently prophesying one's own death might have been terrifying to a prophet, (though things like this happened many times in Scripture: Pharaoh's prophesy about the first born comes to mind, and Proverbs 18:21). There is no reason to conclude that Obadiah's prediction was wrong.