We all know that the kosher laws forbid mixing milk and meat. But what's the proof?
Issi b. Judah said: How do we know that meat and milk [seethed together] are forbidden?
Well, yes; it's more of a problem than you might think. The biblical prohibition says simply: You shall not seethe a kid in its mother’s milk (Deut. 14:21. The same phrase appears with no more explanation in Ex. 23:19 and 34:27).
How do we know that it cannot be eaten – say if someone else does the cooking?
It is stated here, for thou art a holy people [...thou shalt not seethe a kid in its mother's milk], and it is stated elsewhere, And ye shall be holy men unto me; [therefore ye shall not eat any flesh that is torn of beasts in the field; ye shall cast it to the dogs]: (Ex.22:30) just as there it is forbidden, so here too it is forbidden.
By connecting two verses with the common phrase of ‘being holy’ (even though the first is am kadosh and the second is anshei kadosh), we learn that the rule of one applies to the other. Since you cannot eat an animal torn by a wild animal, so you cannot eat the meat which has been seethed in milk!
No mention here of two sets of dishes. That comes elsewhere.