Step | Hyp | Ref
| Expression |
1 | | nbgr2vtx1edg.v |
. . . . 5
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | fvex 6113 |
. . . . 5
⊢
(Vtx‘𝐺) ∈
V |
3 | 1, 2 | eqeltri 2684 |
. . . 4
⊢ 𝑉 ∈ V |
4 | | hash2prb 13111 |
. . . 4
⊢ (𝑉 ∈ V → ((#‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))) |
5 | 3, 4 | ax-mp 5 |
. . 3
⊢
((#‘𝑉) = 2
↔ ∃𝑎 ∈
𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) |
6 | | simpll 786 |
. . . . . . . . . 10
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
7 | 6 | ancomd 466 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉)) |
8 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏) |
9 | 8 | necomd 2837 |
. . . . . . . . . 10
⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → 𝑏 ≠ 𝑎) |
10 | 9 | ad2antlr 759 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ≠ 𝑎) |
11 | | id 22 |
. . . . . . . . . . 11
⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸) |
12 | | sseq2 3590 |
. . . . . . . . . . . 12
⊢ (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏})) |
13 | 12 | adantl 481 |
. . . . . . . . . . 11
⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑎, 𝑏})) |
14 | | ssid 3587 |
. . . . . . . . . . . 12
⊢ {𝑎, 𝑏} ⊆ {𝑎, 𝑏} |
15 | 14 | a1i 11 |
. . . . . . . . . . 11
⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑎, 𝑏}) |
16 | 11, 13, 15 | rspcedvd 3289 |
. . . . . . . . . 10
⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒) |
17 | 16 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒) |
18 | 1 | 1vgrex 25679 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ 𝑉 → 𝐺 ∈ V) |
19 | | nbgr2vtx1edg.e |
. . . . . . . . . . . 12
⊢ 𝐸 = (Edg‘𝐺) |
20 | 1, 19 | nbgrel 40564 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ V → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))) |
21 | 18, 20 | syl 17 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝑉 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))) |
22 | 21 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ≠ 𝑎 ∧ ∃𝑒 ∈ 𝐸 {𝑎, 𝑏} ⊆ 𝑒))) |
23 | 7, 10, 17, 22 | mpbir3and 1238 |
. . . . . . . 8
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) |
24 | 8 | ad2antlr 759 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ≠ 𝑏) |
25 | | sseq2 3590 |
. . . . . . . . . . . 12
⊢ (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏})) |
26 | 25 | adantl 481 |
. . . . . . . . . . 11
⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏})) |
27 | | prcom 4211 |
. . . . . . . . . . . . 13
⊢ {𝑏, 𝑎} = {𝑎, 𝑏} |
28 | 27 | eqimssi 3622 |
. . . . . . . . . . . 12
⊢ {𝑏, 𝑎} ⊆ {𝑎, 𝑏} |
29 | 28 | a1i 11 |
. . . . . . . . . . 11
⊢ ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏}) |
30 | 11, 26, 29 | rspcedvd 3289 |
. . . . . . . . . 10
⊢ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) |
31 | 30 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒) |
32 | 1, 19 | nbgrel 40564 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ V → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))) |
33 | 18, 32 | syl 17 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝑉 → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))) |
34 | 33 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏 ∧ ∃𝑒 ∈ 𝐸 {𝑏, 𝑎} ⊆ 𝑒))) |
35 | 6, 24, 31, 34 | mpbir3and 1238 |
. . . . . . . 8
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) |
36 | | difprsn1 4271 |
. . . . . . . . . . . . 13
⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏}) |
37 | 36 | raleqdv 3121 |
. . . . . . . . . . . 12
⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎))) |
38 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑏 ∈ V |
39 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))) |
40 | 38, 39 | ralsn 4169 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
{𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) |
41 | 37, 40 | syl6bb 275 |
. . . . . . . . . . 11
⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))) |
42 | | difprsn2 4272 |
. . . . . . . . . . . . 13
⊢ (𝑎 ≠ 𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎}) |
43 | 42 | raleqdv 3121 |
. . . . . . . . . . . 12
⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏))) |
44 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑎 ∈ V |
45 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))) |
46 | 44, 45 | ralsn 4169 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
{𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) |
47 | 43, 46 | syl6bb 275 |
. . . . . . . . . . 11
⊢ (𝑎 ≠ 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))) |
48 | 41, 47 | anbi12d 743 |
. . . . . . . . . 10
⊢ (𝑎 ≠ 𝑏 → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))) |
49 | 48 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))) |
50 | 49 | ad2antlr 759 |
. . . . . . . 8
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))) |
51 | 23, 35, 50 | mpbir2and 959 |
. . . . . . 7
⊢ ((((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))) |
52 | 51 | ex 449 |
. . . . . 6
⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))) |
53 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)) |
54 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏}) |
55 | | difeq1 3683 |
. . . . . . . . . . . 12
⊢ (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣})) |
56 | 55 | raleqdv 3121 |
. . . . . . . . . . 11
⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
57 | 54, 56 | raleqbidv 3129 |
. . . . . . . . . 10
⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
58 | | sneq 4135 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 𝑎 → {𝑣} = {𝑎}) |
59 | 58 | difeq2d 3690 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎})) |
60 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎)) |
61 | 60 | eleq2d 2673 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎))) |
62 | 59, 61 | raleqbidv 3129 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎))) |
63 | | sneq 4135 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏}) |
64 | 63 | difeq2d 3690 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏})) |
65 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏)) |
66 | 65 | eleq2d 2673 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏))) |
67 | 64, 66 | raleqbidv 3129 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))) |
68 | 44, 38, 62, 67 | ralpr 4185 |
. . . . . . . . . 10
⊢
(∀𝑣 ∈
{𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))) |
69 | 57, 68 | syl6bb 275 |
. . . . . . . . 9
⊢ (𝑉 = {𝑎, 𝑏} → (∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))) |
70 | 53, 69 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑉 = {𝑎, 𝑏} → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))) |
71 | 70 | adantl 481 |
. . . . . . 7
⊢ ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))) |
72 | 71 | adantl 481 |
. . . . . 6
⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → ((𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏))))) |
73 | 52, 72 | mpbird 246 |
. . . . 5
⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
74 | 73 | ex 449 |
. . . 4
⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))) |
75 | 74 | rexlimivv 3018 |
. . 3
⊢
(∃𝑎 ∈
𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}) → (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
76 | 5, 75 | sylbi 206 |
. 2
⊢
((#‘𝑉) = 2
→ (𝑉 ∈ 𝐸 → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
77 | 76 | imp 444 |
1
⊢
(((#‘𝑉) = 2
∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) |