Children of the Promise

Romans 9 to 11 proves the broken off branches who remain in unbelief are not true Israelites:But it is not as though the word of God has failed. For not all who are descended from Israel belong to Israel, and not all are children of Abraham because they are his offspring, but “Through Isaac shall your offspring be named”.

This means that it is not the ethnic children  who are God’s people, but the children of the promise, who have faith in Him, who are counted as Abrahams descendants.
The statement “For not all who are descended from Israel belong to Israel” clearly denotes that some are ethnic Israel, the remnant of verse 29. But the rest are shown as those who sought righteousness by attempting to keep the Law; verses 30-33. 
As indeed He says in Hosea, “Those who were not my people I will call ‘my people,’ and her who was not beloved I will call ‘beloved.’ And in the very place where it was said to them, ‘You are not my people,’ there they will be called ‘sons of the living God.’
Hosea prophesied of the coming time the descendants of the ten northern tribes, the nation of Israel/Ephraim, would be restored as God’s covenant people once more, which is precisely what the inauguration of the New Covenant accomplishes.

Hebrews 8:7-8 For if that first covenant had been faultless, there would have been no occasion to look for a second. For he finds fault with them when he says: “Behold, the days are coming, declares the Lord, when I will establish a new covenant with the house of Israel and with the house of Judah
The inauguration of the new covenant was accomplished by the blood of Christ, which also released Judah and the House of Israel, the ten Northern tribes from the marriage contract of the Old Covenant. Romans 7:1-4

While Judah, for the most part, rejected Christ, Peter testifies of his ordained success with the elect exiles of the dispersion, Ephraim/Israel.

1 Peter 1:1-12, an apostle of Jesus Christ, To those who are elect exiles of the dispersion in Pontus, Galatia, Cappadocia and Bithynia, chosen in the foreknowledge of God the Father, in the sanctification of the Spirit, for obedience to Jesus Christ and for sprinkling with his blood: May grace and peace be multiplied to you.
Paul, the Apostle to the people called Gentiles, but who are actually the scattered dispersion of Israel, John 7:35, reveals they are fellow heirs in Christ, part of the biological descendants of Abraham. The offspring or ‘seed’, by which the gentiles were prophesied to be blessed, was and is being fulfilled by the descendants of Ephraim/Israel of which Zechariah prophesied:

Zechariah 10:7-12          And they of Ephraim shall be like a mighty man, and their heart shall rejoice as through wine: yes, their children shall see it, and be glad; their hearts shall rejoice in the Lord. I will call for them, and gather them; for I have redeemed them: and they shall increase as they have increased. And I will sow them among the people: and they shall remember me in far countries; and they shall rear their children, then march into and occupy the holy land.

What happened: God called Ephraim and gathered them in Christ and then scattered them in the world to proclaim the Gospel to the nations, which Judah was obviously not part of. The plan for Ephraim is the source of the parable of the wheat and the tares in Matthew 13. Proved by how the Western peoples have sent out missionaries and translated the Bible into most languages.

Isaiah 49:1-26       Listen to Me you peoples far distant,…… But Zion [the Land] said: The Lord has forsaken me, and my Lord has forgotten me…. The children which you will have, after you have lost the former ones, shall say to you: This place is too small for us, give us more space to dwell . Then you will say to yourself: Who has begotten me these, seeing I have lost my children, and am alone and desolate, and where have they come from? Thus says the Lord GOD, Behold, I will lift up mine hand to the Gentiles, and set up my standard to the people: and they shall bring thy sons in their arms, and thy daughters shall be carried upon their shoulders, for they shall not be ashamed that have faith in Me.

Hebrews 12:22-24 But you have come unto mount Zion……
We Christians are the latter children of Zion, of Ephraim, the true Israelites.