Why is the minimum energy equal to the energy uncertainty?

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TL;DR Summary

Why is the minimum energy equal to the energy uncertainty?

I was watching this video on Youtube, however, I don't get the step at 14:50 where he says that ΔE≥½hf means that E₀=½hf.

Could someone explain why the minimum energy is equal to the energy uncertainty?

 

[Vanadium 50]

"Watch this obviously confusing video and then explain it to me" is a big, big ask.

Could someone explain why the minimum energy is equal to the energy uncertainty?

It's not.

 

[Vanadium 50]

Amended: "Watch this obviously confusing 37 minute video..."

 

[PeterDonis]

I don't get the step at 14:50 where he says that ΔE≥½hf means that E0=½hf.

It's just saying that, since the energy of any state whatever must be greater than or equal to $hf / 2$, the energy of the lowest energy state, the ground state energy $E_0$, is equal to $hf / 2$.

Could someone explain why the minimum energy is equal to the energy uncertainty?

That's not what the above is saying.