Step |
Hyp |
Ref |
Expression |
1 |
|
ax-1 |
|- ( N e. NN -> ( X e. ( FPPr ` N ) -> N e. NN ) ) |
2 |
|
df-fppr |
|- FPPr = ( n e. NN |-> { x e. ( ZZ>= ` 4 ) | ( x e/ Prime /\ x || ( ( n ^ ( x - 1 ) ) - 1 ) ) } ) |
3 |
2
|
fvmptndm |
|- ( -. N e. NN -> ( FPPr ` N ) = (/) ) |
4 |
|
eleq2 |
|- ( ( FPPr ` N ) = (/) -> ( X e. ( FPPr ` N ) <-> X e. (/) ) ) |
5 |
|
noel |
|- -. X e. (/) |
6 |
5
|
pm2.21i |
|- ( X e. (/) -> N e. NN ) |
7 |
4 6
|
syl6bi |
|- ( ( FPPr ` N ) = (/) -> ( X e. ( FPPr ` N ) -> N e. NN ) ) |
8 |
3 7
|
syl |
|- ( -. N e. NN -> ( X e. ( FPPr ` N ) -> N e. NN ) ) |
9 |
1 8
|
pm2.61i |
|- ( X e. ( FPPr ` N ) -> N e. NN ) |