Step |
Hyp |
Ref |
Expression |
1 |
|
1nn0 |
|- 1 e. NN0 |
2 |
|
3nn0 |
|- 3 e. NN0 |
3 |
|
1le3 |
|- 1 <_ 3 |
4 |
|
elfz2nn0 |
|- ( 1 e. ( 0 ... 3 ) <-> ( 1 e. NN0 /\ 3 e. NN0 /\ 1 <_ 3 ) ) |
5 |
1 2 3 4
|
mpbir3an |
|- 1 e. ( 0 ... 3 ) |
6 |
|
fzsplit |
|- ( 1 e. ( 0 ... 3 ) -> ( 0 ... 3 ) = ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) ) ) |
7 |
5 6
|
ax-mp |
|- ( 0 ... 3 ) = ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) ) |
8 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
9 |
8
|
oveq2i |
|- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) ) |
10 |
|
0z |
|- 0 e. ZZ |
11 |
|
fzpr |
|- ( 0 e. ZZ -> ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } ) |
12 |
10 11
|
ax-mp |
|- ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } |
13 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
14 |
13
|
preq2i |
|- { 0 , ( 0 + 1 ) } = { 0 , 1 } |
15 |
9 12 14
|
3eqtri |
|- ( 0 ... 1 ) = { 0 , 1 } |
16 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
17 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
18 |
16 17
|
oveq12i |
|- ( ( 1 + 1 ) ... 3 ) = ( 2 ... ( 2 + 1 ) ) |
19 |
|
2z |
|- 2 e. ZZ |
20 |
|
fzpr |
|- ( 2 e. ZZ -> ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } ) |
21 |
19 20
|
ax-mp |
|- ( 2 ... ( 2 + 1 ) ) = { 2 , ( 2 + 1 ) } |
22 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
23 |
22
|
preq2i |
|- { 2 , ( 2 + 1 ) } = { 2 , 3 } |
24 |
18 21 23
|
3eqtri |
|- ( ( 1 + 1 ) ... 3 ) = { 2 , 3 } |
25 |
15 24
|
uneq12i |
|- ( ( 0 ... 1 ) u. ( ( 1 + 1 ) ... 3 ) ) = ( { 0 , 1 } u. { 2 , 3 } ) |
26 |
7 25
|
eqtri |
|- ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } ) |