Proof of Theorem edgnbusgreu
Step | Hyp | Ref
| Expression |
1 | | simpll 767 |
. . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝐺 ∈ USGraph) |
2 | | edgnbusgreu.e |
. . . . . . . 8
⊢ 𝐸 = (Edg‘𝐺) |
3 | 2 | eleq2i 2824 |
. . . . . . 7
⊢ (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ (Edg‘𝐺)) |
4 | 3 | biimpi 219 |
. . . . . 6
⊢ (𝐶 ∈ 𝐸 → 𝐶 ∈ (Edg‘𝐺)) |
5 | 4 | ad2antrl 728 |
. . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝐶 ∈ (Edg‘𝐺)) |
6 | | simprr 773 |
. . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → 𝑀 ∈ 𝐶) |
7 | | usgredg2vtxeu 27155 |
. . . . 5
⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ∈ (Edg‘𝐺) ∧ 𝑀 ∈ 𝐶) → ∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛}) |
8 | 1, 5, 6, 7 | syl3anc 1372 |
. . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛}) |
9 | | df-reu 3060 |
. . . . 5
⊢
(∃!𝑛 ∈
(Vtx‘𝐺)𝐶 = {𝑀, 𝑛} ↔ ∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) |
10 | | prcom 4620 |
. . . . . . . . . . . . . . . 16
⊢ {𝑀, 𝑛} = {𝑛, 𝑀} |
11 | 10 | eqeq2i 2751 |
. . . . . . . . . . . . . . 15
⊢ (𝐶 = {𝑀, 𝑛} ↔ 𝐶 = {𝑛, 𝑀}) |
12 | 11 | biimpi 219 |
. . . . . . . . . . . . . 14
⊢ (𝐶 = {𝑀, 𝑛} → 𝐶 = {𝑛, 𝑀}) |
13 | 12 | eleq1d 2817 |
. . . . . . . . . . . . 13
⊢ (𝐶 = {𝑀, 𝑛} → (𝐶 ∈ 𝐸 ↔ {𝑛, 𝑀} ∈ 𝐸)) |
14 | 13 | biimpcd 252 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ 𝐸 → (𝐶 = {𝑀, 𝑛} → {𝑛, 𝑀} ∈ 𝐸)) |
15 | 14 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (𝐶 = {𝑀, 𝑛} → {𝑛, 𝑀} ∈ 𝐸)) |
16 | 15 | adantld 494 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) → {𝑛, 𝑀} ∈ 𝐸)) |
17 | 16 | imp 410 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → {𝑛, 𝑀} ∈ 𝐸) |
18 | | simprr 773 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → 𝐶 = {𝑀, 𝑛}) |
19 | 17, 18 | jca 515 |
. . . . . . . 8
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) → ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) |
20 | | simpl 486 |
. . . . . . . . . 10
⊢ (({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}) → {𝑛, 𝑀} ∈ 𝐸) |
21 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
22 | 2, 21 | usgrpredgv 27131 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑀} ∈ 𝐸) → (𝑛 ∈ (Vtx‘𝐺) ∧ 𝑀 ∈ (Vtx‘𝐺))) |
23 | 22 | simpld 498 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ USGraph ∧ {𝑛, 𝑀} ∈ 𝐸) → 𝑛 ∈ (Vtx‘𝐺)) |
24 | 1, 20, 23 | syl2an 599 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → 𝑛 ∈ (Vtx‘𝐺)) |
25 | | simprr 773 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → 𝐶 = {𝑀, 𝑛}) |
26 | 24, 25 | jca 515 |
. . . . . . . 8
⊢ ((((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) ∧ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) → (𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛})) |
27 | 19, 26 | impbida 801 |
. . . . . . 7
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
28 | 27 | eubidv 2587 |
. . . . . 6
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) ↔ ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
29 | 28 | biimpd 232 |
. . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ (Vtx‘𝐺) ∧ 𝐶 = {𝑀, 𝑛}) → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
30 | 9, 29 | syl5bi 245 |
. . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛 ∈ (Vtx‘𝐺)𝐶 = {𝑀, 𝑛} → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
31 | 8, 30 | mpd 15 |
. . 3
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛})) |
32 | | edgnbusgreu.n |
. . . . . . . 8
⊢ 𝑁 = (𝐺 NeighbVtx 𝑀) |
33 | 32 | eleq2i 2824 |
. . . . . . 7
⊢ (𝑛 ∈ 𝑁 ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑀)) |
34 | 2 | nbusgreledg 27287 |
. . . . . . 7
⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑀) ↔ {𝑛, 𝑀} ∈ 𝐸)) |
35 | 33, 34 | syl5bb 286 |
. . . . . 6
⊢ (𝐺 ∈ USGraph → (𝑛 ∈ 𝑁 ↔ {𝑛, 𝑀} ∈ 𝐸)) |
36 | 35 | anbi1d 633 |
. . . . 5
⊢ (𝐺 ∈ USGraph → ((𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
37 | 36 | ad2antrr 726 |
. . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ((𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
38 | 37 | eubidv 2587 |
. . 3
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → (∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛}) ↔ ∃!𝑛({𝑛, 𝑀} ∈ 𝐸 ∧ 𝐶 = {𝑀, 𝑛}))) |
39 | 31, 38 | mpbird 260 |
. 2
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛})) |
40 | | df-reu 3060 |
. 2
⊢
(∃!𝑛 ∈
𝑁 𝐶 = {𝑀, 𝑛} ↔ ∃!𝑛(𝑛 ∈ 𝑁 ∧ 𝐶 = {𝑀, 𝑛})) |
41 | 39, 40 | sylibr 237 |
1
⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ 𝑁 𝐶 = {𝑀, 𝑛}) |