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no it's not supposed to happen in 2040, i think you are talking about the date of ending of block issuance, which is 2140 & not 2040..
however, the mining won't stop then, i think it will cost more for transactions and that's it..

Yeah it wont happen for a long time, but the starting effects of going towards zero will be seen heavily with in the next 2-3 halvings which isnt that far away.

I may hope that by that time Bitcoin is outpassed in its market cap ^^