Pushy, 8 bytes

Simple implementation of checking whether len(set(list)) == len(list)-1:

LtvuL^=#

Explanation:

       \ Implicit: Put all input on stack
Ltv    \ Get the stack length - 1, save in auxiliary stack
u      \ Remove non-unique elements
L      \ Get the new length
^=     \ Compare with the previously saved length
#      \ Print result

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Pushy, 8 bytes

Simple implementation of checking whether len(set(list)) == len(list)-1:

LtvuL^=#

Explanation:

       \ Implicit: Put all input on stack
Ltv    \ Get the stack length - 1, save in auxiliary stack
u      \ Remove non-unique elements
L      \ Get the new length
^=     \ Compare with the previously saved length
#      \ Print result

This works as the length will only decrease by 1 if there was only exactly 1 non-distinct integer in the initial list.

Try it online!