Step |
Hyp |
Ref |
Expression |
1 |
|
stgredgel |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑒 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } ) ) ) |
2 |
|
eliun |
⊢ ( 𝑒 ∈ ∪ 𝑥 ∈ ( 1 ... 𝑁 ) { { 0 , 𝑥 } } ↔ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 ∈ { { 0 , 𝑥 } } ) |
3 |
2
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑒 ∈ ∪ 𝑥 ∈ ( 1 ... 𝑁 ) { { 0 , 𝑥 } } ↔ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 ∈ { { 0 , 𝑥 } } ) ) |
4 |
|
velsn |
⊢ ( 𝑒 ∈ { { 0 , 𝑥 } } ↔ 𝑒 = { 0 , 𝑥 } ) |
5 |
|
0elfz |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → 0 ∈ ( 0 ... 𝑁 ) ) |
7 |
|
fz1ssfz0 |
⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 ) |
8 |
7
|
sseli |
⊢ ( 𝑥 ∈ ( 1 ... 𝑁 ) → 𝑥 ∈ ( 0 ... 𝑁 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → 𝑥 ∈ ( 0 ... 𝑁 ) ) |
10 |
6 9
|
prssd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) ) |
11 |
|
sseq1 |
⊢ ( 𝑒 = { 0 , 𝑥 } → ( 𝑒 ⊆ ( 0 ... 𝑁 ) ↔ { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) ) ) |
12 |
10 11
|
syl5ibrcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( 𝑒 = { 0 , 𝑥 } → 𝑒 ⊆ ( 0 ... 𝑁 ) ) ) |
13 |
12
|
pm4.71rd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( 𝑒 = { 0 , 𝑥 } ↔ ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ 𝑒 = { 0 , 𝑥 } ) ) ) |
14 |
4 13
|
bitr2id |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ 𝑒 = { 0 , 𝑥 } ) ↔ 𝑒 ∈ { { 0 , 𝑥 } } ) ) |
15 |
14
|
rexbidva |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑥 ∈ ( 1 ... 𝑁 ) ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ 𝑒 = { 0 , 𝑥 } ) ↔ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 ∈ { { 0 , 𝑥 } } ) ) |
16 |
|
r19.42v |
⊢ ( ∃ 𝑥 ∈ ( 1 ... 𝑁 ) ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ 𝑒 = { 0 , 𝑥 } ) ↔ ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } ) ) |
17 |
16
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑥 ∈ ( 1 ... 𝑁 ) ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ 𝑒 = { 0 , 𝑥 } ) ↔ ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } ) ) ) |
18 |
3 15 17
|
3bitr2rd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑒 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } ) ↔ 𝑒 ∈ ∪ 𝑥 ∈ ( 1 ... 𝑁 ) { { 0 , 𝑥 } } ) ) |
19 |
1 18
|
bitrd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑒 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ 𝑒 ∈ ∪ 𝑥 ∈ ( 1 ... 𝑁 ) { { 0 , 𝑥 } } ) ) |
20 |
19
|
eqrdv |
⊢ ( 𝑁 ∈ ℕ0 → ( Edg ‘ ( StarGr ‘ 𝑁 ) ) = ∪ 𝑥 ∈ ( 1 ... 𝑁 ) { { 0 , 𝑥 } } ) |