Biblical source of the seven Noahide laws

What's the Biblical source of the seven noahide laws?

1 "Va'Ytzav" - this refers to Dinim, as it says "Asher Yetzaveh Es Banav (...La'asos Tzedakah u'Mishpat)"  
2 "Hash-m" refers to blasphemy - "V'Nokev Shem Hash-m..."
3 "Elokim" refers to idolatry - "Lo Yihyeh Lecha Elokim Acherim" 
4 "Leimor" refers to Arayos - "Leimor Hen Yeshalach Ish Es Ishto...v'Haysah l'Ish Acher" 
5 "Al ha'Adam" refers to murder - "Shofech Dam ha'Adam" 
6 "Mi'Kol Etz ha'Gan" - not from theft 
7 "Achol Tochal" - not a limb of a living animal

 

Bereshit 2:16

  Va'Ytzav Hash-m Elokim Al ha'Adam Leimor Mi'Kol Etz ha'Gan Achol Tochal
 
Genesis 2:16 And the LORD God commanded the man, saying: 'Of every tree of the garden thou mayest freely eat'

 

From The Kuzari (3:73)

 "God, the Almighty commanded Adam saying, 'From all the trees of the garden you may eat'" (Bereishis, 2:16) is an allusion to the seven Noachide commandments.

 The interpretation of this verse is as follows:

"commanded" refers to laws (concerning monetary matters),

"God" refers to blasphemy,

"Almighty" refers to idolatry,

"Adam" refers to murder,

"saying" refers to adultery,

"from all the trees of the garden" refers to thievery,

"you may eat" refers to eating the flesh of a live animal.

 {It is quite obvious} that the verse has little connection with the above laws. However, the Sages intended {their references} as a hint which helps one remember something which the entire nation knows—the seven Noachide commandments.

 

From Talmud Bavli Sanhedrin 56b

(English translation by Soncino)

Whence do we know this? — R. Johanan answered: The Writ saith: And the Lord God commanded the man saying, of every tree of the garden thou mayest freely eat. 1  And [He] commanded, refers to [the observance of] social laws, and thus it is written, For I know him, that he will command his children and his household after him, and they shall keep the way of the Lord, to do justice and judgment. 2  The Lord — is [a prohibition against] blasphemy, and thus it is written, and he that blasphemeth the name of the Lord, he shall surely be put to death. 3   God — is [an injunction against] idolatry, and thus it is written, Thou shalt have no other gods before Me. 4 The man — refers to bloodshed [murder], and thus it is written, Whoso sheddeth man's blood, by man shall his blood be shed. 5   Saying — refers to adultery, and thus it is written, They say, If a man put away his wife, and she go from him, and became another man's. 6  Of every tree of the garden — but not of robbery. 7  Thou mayest freely eat — but not flesh cut from a living animal. 8

notes

Gen. 2:16

Gen. 18:19. Thus 'command' relates to justice and judgment.

Lev. 24:16 — 'The Lord' being used in connection with blasphemy.

Ex. 20:3.

Gen. 9: 6.

Jer. 3:1. Thus 'saying' is used in connection with adultery.

Since it was necessary to authorize Adam to eat of the trees of the garden, it follows that without such authorisation — i.e., when something belongs to another — it is forbidden.

By interpreting thus: Thou mayest eat that which is now ready for eating, but not whilst the animal is alive. It is perhaps remarkable that a verse, the literal meaning of which is obviously permission to enjoy, should be interpreted as a series of prohibitions. Yet it is quite in keeping with the character of the Talmud: freedom to enjoy must be limited by moral and social considerations, and indeed only attains its highest value when so limited. Cf. Ab. VI, 2: No man is free but he who labours in the Torah.