Revision 0760db56-455a-4952-aab5-bc4e47ba7b93 - Code Golf Stack Exchange

# [Retina 0.8.2], 53 bytes

 ^
 ;
 +%`;(.)
 1ドル;$'¶$`;
 .+
 $*
 %A`^.?$|^(..+)1円+$
 ^1+¶+$

[Try it online!][TIO-kfbkobs1] Link includes test cases. Explanation:

 ^
 ;
 +%`;(.)
 1ドル;$'¶$`;

Generate all subsequences of the input.

 .+
 $*

Convert then to unary.

 %A`^.?$|^(..+)1円+$

Delete the ones that aren't prime, but don't delete the newlines. (A multiline replace also works, but is harder to format an explanation for.)

 ^1+¶+$

Check that the original input was prime but none of the proper subsequences were.

[Retina 0.8.2]: https://github.com/m-ender/retina/wiki/The-Language/a950ad7d925ec9316e3e2fb2cf5d49fd15d23e3d
[TIO-kfbkobs1]: https://tio.run/##DcO7DYAgEADQ/uY4InjJRRQ/hMJYuYQhWFjYWBhL53IAFsN7ybuP57z2ovSaSoQApFLQbABtwCp/mAIwAdaglhR5xjdqZjKbJYRoKX@EpbjGg7OylZ10speDHOUkvf8B "Retina 0.8.2 – Try It Online"