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Retina 0.8.2, 53 bytes

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

Try it online! 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.