To assume that the Left is becoming extinct would be a very bad idea, and it would be allowing hubris to control your actions.
The country is evenly split, and the Left still has plenty of influence.
If Trump kicks ass and does well, great, good for America. But it's on him and the GOP now.
It won't take much for the GOP to be on their backs again, so over-interpreting this "mandate" would be foolish.
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It's not "evenly split", with only slightly more than 20% of the adult populace self-identifying as "liberal" and nearly 40% as "conservative", if the Democrats as a party lurch farther to the left the left will indeed risk becoming "extinct" from a political power standpoint since they'll have shrunk their only viable vehicle to power into a regional party.
Fortunately for the Democrats I don't believe that the Democrat Powers that be are going to let that happen, they'll throw the left wingers some crumbs and find another pseudo centrist Clinton look a like to carry the party standard, Obama was the lefty wet dream and just look at what his tenure did to the political power of the left.
Liberals are less likely to identify as liberal, many think of themselves as "mainstream". Conservatives are more likely to be proud and open about their ideology.
Upon what evidence do you base that assertion? You're essentially saying that left wingers are ashamed to self identify as "liberals" and self identify as moderates instead, which I have never seen any evidence to support.
Just look at the popular vote. This is a split country.
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Yes just look at the popular vote and where Hillary Clinton got most of hers, then look at the other 80% of the Republic, clearly the conservatives and the moderates dominate the landscape interrupted only by coastal left wing enclaves.
This country is split from a Democrat/Republican standpoint not from a left wing/right wing standpoint, the decimation of the Democrats under the current left winger in the White House should demonstrate that, if the Democrats go farther to the left, they'll end up being a smaller regional party than they are right now and the left wingers will be left entirely out in the cold from a national political power standpoint since they need moderates more than conservatives do.