Step |
Hyp |
Ref |
Expression |
1 |
|
cplgredgex.1 |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
cplgredgex.2 |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
simp2 |
⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → 𝐴 ∈ 𝑉 ) |
4 |
|
simp3 |
⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) |
5 |
|
eleq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ∈ 𝑉 ↔ 𝐴 ∈ 𝑉 ) ) |
6 |
|
sneq |
⊢ ( 𝑎 = 𝐴 → { 𝑎 } = { 𝐴 } ) |
7 |
6
|
difeq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑉 ∖ { 𝑎 } ) = ( 𝑉 ∖ { 𝐴 } ) ) |
8 |
7
|
eleq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ↔ 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ) ) |
9 |
5 8
|
anbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) ↔ ( 𝐴 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ) ) ) |
10 |
|
preq1 |
⊢ ( 𝑎 = 𝐴 → { 𝑎 , 𝑏 } = { 𝐴 , 𝑏 } ) |
11 |
10
|
sseq1d |
⊢ ( 𝑎 = 𝐴 → ( { 𝑎 , 𝑏 } ⊆ 𝑒 ↔ { 𝐴 , 𝑏 } ⊆ 𝑒 ) ) |
12 |
11
|
rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝑏 } ⊆ 𝑒 ) ) |
13 |
9 12
|
imbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ↔ ( ( 𝐴 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝑏 } ⊆ 𝑒 ) ) ) |
14 |
|
eleq1 |
⊢ ( 𝑏 = 𝐵 → ( 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ↔ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) ) |
15 |
14
|
anbi2d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝐴 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ) ↔ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) ) ) |
16 |
|
preq2 |
⊢ ( 𝑏 = 𝐵 → { 𝐴 , 𝑏 } = { 𝐴 , 𝐵 } ) |
17 |
16
|
sseq1d |
⊢ ( 𝑏 = 𝐵 → ( { 𝐴 , 𝑏 } ⊆ 𝑒 ↔ { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) |
18 |
17
|
rexbidv |
⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝑏 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) |
19 |
15 18
|
imbi12d |
⊢ ( 𝑏 = 𝐵 → ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝑏 } ⊆ 𝑒 ) ↔ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) ) |
20 |
13 19
|
sylan9bb |
⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ↔ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) ) |
21 |
1 2
|
iscplgredg |
⊢ ( 𝐺 ∈ ComplGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ) |
22 |
21
|
ibi |
⊢ ( 𝐺 ∈ ComplGraph → ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) |
23 |
|
rsp2 |
⊢ ( ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ) |
24 |
22 23
|
syl |
⊢ ( 𝐺 ∈ ComplGraph → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ) |
25 |
24
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ ( 𝑉 ∖ { 𝑎 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ) |
26 |
3 4 20 25
|
vtocl2d |
⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) |
27 |
3 4 26
|
mp2and |
⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) |
28 |
27
|
3expib |
⊢ ( 𝐺 ∈ ComplGraph → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ( 𝑉 ∖ { 𝐴 } ) ) → ∃ 𝑒 ∈ 𝐸 { 𝐴 , 𝐵 } ⊆ 𝑒 ) ) |