Step |
Hyp |
Ref |
Expression |
1 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
2 |
|
stgrfv |
⊢ ( 1 ∈ ℕ0 → ( StarGr ‘ 1 ) = { 〈 ( Base ‘ ndx ) , ( 0 ... 1 ) 〉 , 〈 ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) 〉 } ) |
3 |
1 2
|
ax-mp |
⊢ ( StarGr ‘ 1 ) = { 〈 ( Base ‘ ndx ) , ( 0 ... 1 ) 〉 , 〈 ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) 〉 } |
4 |
|
fz01pr |
⊢ ( 0 ... 1 ) = { 0 , 1 } |
5 |
4
|
opeq2i |
⊢ 〈 ( Base ‘ ndx ) , ( 0 ... 1 ) 〉 = 〈 ( Base ‘ ndx ) , { 0 , 1 } 〉 |
6 |
|
elsni |
⊢ ( 𝑥 ∈ { 1 } → 𝑥 = 1 ) |
7 |
|
preq2 |
⊢ ( 𝑥 = 1 → { 0 , 𝑥 } = { 0 , 1 } ) |
8 |
7
|
eqeq2d |
⊢ ( 𝑥 = 1 → ( 𝑒 = { 0 , 𝑥 } ↔ 𝑒 = { 0 , 1 } ) ) |
9 |
8
|
biimpd |
⊢ ( 𝑥 = 1 → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) ) |
10 |
6 9
|
syl |
⊢ ( 𝑥 ∈ { 1 } → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) ) |
11 |
|
1z |
⊢ 1 ∈ ℤ |
12 |
|
fzsn |
⊢ ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } ) |
13 |
11 12
|
ax-mp |
⊢ ( 1 ... 1 ) = { 1 } |
14 |
10 13
|
eleq2s |
⊢ ( 𝑥 ∈ ( 1 ... 1 ) → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) ) |
15 |
14
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) |
16 |
15
|
adantl |
⊢ ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) → 𝑒 = { 0 , 1 } ) |
17 |
|
c0ex |
⊢ 0 ∈ V |
18 |
17
|
prid1 |
⊢ 0 ∈ { 0 , 1 } |
19 |
18 4
|
eleqtrri |
⊢ 0 ∈ ( 0 ... 1 ) |
20 |
|
1ex |
⊢ 1 ∈ V |
21 |
20
|
prid2 |
⊢ 1 ∈ { 0 , 1 } |
22 |
21 4
|
eleqtrri |
⊢ 1 ∈ ( 0 ... 1 ) |
23 |
|
prelpwi |
⊢ ( ( 0 ∈ ( 0 ... 1 ) ∧ 1 ∈ ( 0 ... 1 ) ) → { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ) |
24 |
19 22 23
|
mp2an |
⊢ { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) |
25 |
|
eqid |
⊢ { 0 , 1 } = { 0 , 1 } |
26 |
13
|
rexeqi |
⊢ ( ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ↔ ∃ 𝑥 ∈ { 1 } { 0 , 1 } = { 0 , 𝑥 } ) |
27 |
7
|
eqeq2d |
⊢ ( 𝑥 = 1 → ( { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } ) ) |
28 |
20 27
|
rexsn |
⊢ ( ∃ 𝑥 ∈ { 1 } { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } ) |
29 |
26 28
|
bitri |
⊢ ( ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } ) |
30 |
25 29
|
mpbir |
⊢ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } |
31 |
24 30
|
pm3.2i |
⊢ ( { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ) |
32 |
|
eleq1 |
⊢ ( 𝑒 = { 0 , 1 } → ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ↔ { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ) ) |
33 |
|
eqeq1 |
⊢ ( 𝑒 = { 0 , 1 } → ( 𝑒 = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 𝑥 } ) ) |
34 |
33
|
rexbidv |
⊢ ( 𝑒 = { 0 , 1 } → ( ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ↔ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ) ) |
35 |
32 34
|
anbi12d |
⊢ ( 𝑒 = { 0 , 1 } → ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) ↔ ( { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ) ) ) |
36 |
31 35
|
mpbiri |
⊢ ( 𝑒 = { 0 , 1 } → ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) ) |
37 |
16 36
|
impbii |
⊢ ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) ↔ 𝑒 = { 0 , 1 } ) |
38 |
37
|
abbii |
⊢ { 𝑒 ∣ ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) } = { 𝑒 ∣ 𝑒 = { 0 , 1 } } |
39 |
|
df-rab |
⊢ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } = { 𝑒 ∣ ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) } |
40 |
|
df-sn |
⊢ { { 0 , 1 } } = { 𝑒 ∣ 𝑒 = { 0 , 1 } } |
41 |
38 39 40
|
3eqtr4i |
⊢ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } = { { 0 , 1 } } |
42 |
41
|
reseq2i |
⊢ ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) = ( I ↾ { { 0 , 1 } } ) |
43 |
42
|
opeq2i |
⊢ 〈 ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) 〉 = 〈 ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) 〉 |
44 |
5 43
|
preq12i |
⊢ { 〈 ( Base ‘ ndx ) , ( 0 ... 1 ) 〉 , 〈 ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) 〉 } = { 〈 ( Base ‘ ndx ) , { 0 , 1 } 〉 , 〈 ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) 〉 } |
45 |
3 44
|
eqtri |
⊢ ( StarGr ‘ 1 ) = { 〈 ( Base ‘ ndx ) , { 0 , 1 } 〉 , 〈 ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) 〉 } |