Cup
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Atonement

“The manner of administering the wine–Behold, they took the cup, and said: O God, the Eternal Father, we ask thee, in the name of thy Son, Jesus Christ, to bless and sanctify this wine to the souls of all those who drink of it, that they may do it in remembrance of the blood of thy Son, which was shed for them; that they may witness unto thee, O God, the Eternal Father, that they do always remember him, that they may have his Spirit to be with them. Amen.”
Moroni 5:1-2

“And behold, I am the light and the life of the world; and I have drunk out of that bitter cup which the Father hath given me, and have glorified the Father in taking upon me the sins of the world, in the which I have suffered the will of the Father in all things from the beginning.”
3 Nephi 11:11

“…I, having accomplished and finished the will of him whose I am, even the Father…and would that I might not drink the bitter cup, and shrink— Nevertheless, glory be to the Father, and I partook and finished…”
D&C 19:2,18,19

Betrothal

“When a young man was ready for marriage, he went to the home of his prospective bride, accompanied by his father and a friend. The young lady receiving the guests would have her father, her mother, and usually a friend. After the purchase price of the bride was negotiated and agreed, the young man poured a cup of wine for the young woman and placed it before her. She had a short time to make her decision. If she refused, the young suitor would leave never to return. If she drank of the glass, they were bound by covenant and were now betrothed.”
Richard P. Himmer, The Bitter Cup – Link

Covenants

Drinking the cup may represent a sign that one accepts God’s will, or entrance into or a renewal of a covenant. By taking the contents of the cup into yourself, you internalize what has been presented to you. A cup can be used to receive, transport and give.

“After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me. For as often as ye eat this bread, and drink this cup, ye do shew the Lord’s death till he come.”
1 Corinthians 11:25-26

“Then took I the cup at the Lord’s hand…”
Jeremiah 25:17

“Thou hast walked in the way of thy sister; therefore will I give her cup into thine hand.”
Ezekiel 23:31

Vessel

In LDS theology, a cup is a vessel that typically holds liquid, for example, the sacrament cup holds wine or water. The Father gave Jesus a symbolic cup containing a mission, to atone for the sins of mankind. Our bodies are like vessels that hold our life, our blood and our spirits. In the ancient temple, censers (spoon or cup-like instruments) held frankincense (a typically white, yellow or red hard resin) for burning on the altar before the veil.

“In the Temple, the priest evidently stood with hand in cupping shape, ready to receive something which was given to him. It was probably incense, though, in the last days (see Rev. 2:17; D&C 130:11), it will evidently be the white stone or urim and thummim, with the new name written in it.”
Dr. Lynn M. Hilton, The Hand as a Cup in Ancient Temple Worship – Link

“Likewise also the cup after supper, saying, This cup is the new testament in my blood, which is shed for you.”
Luke 22:20

“After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me.”
1 Corinthians 11:25

See also “Atonement”