1Ki 19:14 And he said, I have been very jealous for the LORD God of hosts: because the children of Israel have forsaken thy covenant, thrown down thine altars, and slain thy prophets with the sword; and I, [even] I only, am left; and they seek my life, to take it away.
1Ki 19:18 Yet I have left [me] seven thousand in Israel, all the knees which have not bowed unto Baal, and every mouth which hath not kissed him.
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Jhn 14:16 And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever;
Jhn 14:17 [Even] the Spirit of truth; whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him; for he dwelleth with you, and shall be in you.
Jhn 14:18 I will not leave you comfortless: I will come to you.
2Ti 2:19 Nevertheless the foundation of God standeth sure, having this seal, The Lord knoweth them that are his. And, Let every one that nameth the name of Christ depart from iniquity.
At noi time has God ever been without believers on the earth.
Here’s a good read on the subject. This is a wikipedia article on the great apostasy. It’s not about the LDS perspective, although, the LDS view is also included.