Often to prove that believers who no longer believe 'unbelieving believers', that they are still saved, is Christ's statement that the "other sheep", them he must bring:
John 10:16 And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd.
John 10:16 καὶ ἄλλα πρόβατα ἔχω ἃ οὐκ ἔστιν ἐκ τῆς αὐλῆς ταύτης· κἀκεῖνα δεῖ με ἀγαγεῖν καὶ τῆς φωνῆς μου ἀκούσουσιν, καὶ γενήσονται μία ποίμνη, εἷς ποιμήν.
Obviously that's a pretty bold verse, that the other sheep must be brought. So the typical stance to this is "well, that's just referring to the Israelites, not unbelieving believers". You may make that mistake if you don't compare all of the "sheep" terms and corresponding verses, here are all of them and a sample verse where they appear:
Sheep (John 10:14, 10:27)
Lost Sheep (Matt 10:6)
Astray Sheep (1 Peter 2:25)
'Other' Sheep (John 10:16)
'Not-a-Sheep' (John 10:26)
So let's take a look at the 'claimed lost sheep of Israel', which are mentioned in Matthew 10:6
Matthew 10:6 But go rather to the lost sheep of the house of Israel.
Matthew 10:6 πορεύεσθε δὲ μᾶλλον πρὸς τὰ πρόβατα τὰ ἀπολωλότα οἴκου Ἰσραήλ.
Now let's see if unbelieving Jews can autonomously equate to being 'other sheep':
John 10:19 There was a division therefore again among the Jews for these sayings.
John 10:19 Σχίσμα πάλιν ἐγένετο ἐν τοῖς Ἰουδαίοις διὰ τοὺς λόγους τούτους.
John 10:26 But you believe not, because you are not of my sheep, as I said to you.
John 10:26 ἀλλὰ ὑμεῖς οὐ πιστεύετε, ὅτι οὐκ ἐστὲ ἐκ τῶν προβάτων τῶν ἐμῶν.
The audience is established of being the Jews (in verse 19), who in this particular passage did not BELIEVE (verse 26); and Christ says as a result you are NOT of my sheep because they did not believe.
Therefore, the "other sheep" are not the "lost sheep of Israel", and in order to be 'of Christ's sheep' you had to have believed
. Obviously it can still be contested by those who believe in earning salvation that the "other sheep" are still the Israelites exclusively. But you'd have to eliminate the context and pretend it's all the same.
More troubling, if you didn't consider that 'other sheep' could apply (re: unbelieving believers), you'd break Luke 8:15:, because then all believers would just *be* sheep and there would be no unique modifiers such as 'other sheep'. Well, this is also why Luke 8:10-15 has four types identified.