Something is antithetical when it is in complete and utter opposition to the character of something. If you’re a vegetarian, eating giant T-bone steaks is antithetical to your beliefs.
You’ll find that antithetical is a useful word when you’re trying to express an extreme contrast. If someone is trying to convince you to do something you don’t believe in, or to allow an organization you’re involved with to adopt measures you are strongly against, you can use antithetical. You go out to vote because sitting out of the voting process is antithetical to what you believe in. If you find that your club wants to adopt exclusive measures when it is supposed to be open, you'll tell them that that's antithetical to the club's purpose.