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Theorem nbuhgr2vtx1edgb 26445
Description: If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
Hypotheses
Ref Expression
nbgr2vtx1edg.v 𝑉 = (Vtx‘𝐺)
nbgr2vtx1edg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbuhgr2vtx1edgb ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Distinct variable groups:   𝑛,𝐸   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣
Allowed substitution hint:   𝐸(𝑣)

Proof of Theorem nbuhgr2vtx1edgb
Dummy variables 𝑎 𝑏 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgr2vtx1edg.v . . . . 5 𝑉 = (Vtx‘𝐺)
21fvexi 6361 . . . 4 𝑉 ∈ V
3 hash2prb 13444 . . . 4 (𝑉 ∈ V → ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
42, 3ax-mp 5 . . 3 ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}))
5 simpr 479 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑉𝑏𝑉))
65ancomd 466 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑉𝑎𝑉))
76ad2antrr 764 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏𝑉𝑎𝑉))
8 id 22 . . . . . . . . . . . . 13 (𝑎𝑏𝑎𝑏)
98necomd 2985 . . . . . . . . . . . 12 (𝑎𝑏𝑏𝑎)
109adantr 472 . . . . . . . . . . 11 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑏𝑎)
1110ad2antlr 765 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏𝑎)
12 prcom 4409 . . . . . . . . . . . . . 14 {𝑎, 𝑏} = {𝑏, 𝑎}
1312eleq1i 2828 . . . . . . . . . . . . 13 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
1413biimpi 206 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
15 sseq2 3766 . . . . . . . . . . . . 13 (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1615adantl 473 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1712eqimssi 3798 . . . . . . . . . . . . 13 {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
1817a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
1914, 16, 18rspcedvd 3454 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
2019adantl 473 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
21 nbgr2vtx1edg.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
221, 21nbgrel 26430 . . . . . . . . . 10 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
237, 11, 20, 22syl3anbrc 1429 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
245ad2antrr 764 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎𝑉𝑏𝑉))
25 simplrl 819 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎𝑏)
26 id 22 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
27 sseq2 3766 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
2827adantl 473 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
29 prcom 4409 . . . . . . . . . . . . . 14 {𝑏, 𝑎} = {𝑎, 𝑏}
3029eqimssi 3798 . . . . . . . . . . . . 13 {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
3130a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
3226, 28, 31rspcedvd 3454 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
3332adantl 473 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
341, 21nbgrel 26430 . . . . . . . . . 10 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
3524, 25, 33, 34syl3anbrc 1429 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
3623, 35jca 555 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
3736ex 449 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
381, 21nbuhgr2vtx1edgblem 26444 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
39383exp 1113 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4039adantr 472 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4140adantld 484 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4241imp 444 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
4342adantld 484 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ((𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸))
4437, 43impbid 202 . . . . . 6 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
45 eleq1 2825 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
4645adantl 473 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47 id 22 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48 difeq1 3862 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
4948raleqdv 3281 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5047, 49raleqbidv 3289 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51 vex 3341 . . . . . . . . . . 11 𝑎 ∈ V
52 vex 3341 . . . . . . . . . . 11 𝑏 ∈ V
53 sneq 4329 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → {𝑣} = {𝑎})
5453difeq2d 3869 . . . . . . . . . . . 12 (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
55 oveq2 6819 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
5655eleq2d 2823 . . . . . . . . . . . 12 (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
5754, 56raleqbidv 3289 . . . . . . . . . . 11 (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
58 sneq 4329 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑣} = {𝑏})
5958difeq2d 3869 . . . . . . . . . . . 12 (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
60 oveq2 6819 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
6160eleq2d 2823 . . . . . . . . . . . 12 (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6259, 61raleqbidv 3289 . . . . . . . . . . 11 (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6351, 52, 57, 62ralpr 4380 . . . . . . . . . 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)))
64 difprsn1 4473 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
6564raleqdv 3281 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
66 eleq1 2825 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
6752, 66ralsn 4364 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
6865, 67syl6bb 276 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
69 difprsn2 4474 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
7069raleqdv 3281 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
71 eleq1 2825 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7251, 71ralsn 4364 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
7370, 72syl6bb 276 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7468, 73anbi12d 749 . . . . . . . . . 10 (𝑎𝑏 → ((∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ∧ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7563, 74syl5bb 272 . . . . . . . . 9 (𝑎𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7650, 75sylan9bbr 739 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7746, 76bibi12d 334 . . . . . . 7 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → ((𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
7877adantl 473 . . . . . 6 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ((𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
7944, 78mpbird 247 . . . . 5 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8079ex 449 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8180rexlimdvva 3174 . . 3 (𝐺 ∈ UHGraph → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
824, 81syl5bi 232 . 2 (𝐺 ∈ UHGraph → ((♯‘𝑉) = 2 → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8382imp 444 1 ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wne 2930  wral 3048  wrex 3049  Vcvv 3338  cdif 3710  wss 3713  {csn 4319  {cpr 4321  cfv 6047  (class class class)co 6811  2c2 11260  chash 13309  Vtxcvtx 26071  Edgcedg 26136  UHGraphcuhgr 26148   NeighbVtx cnbgr 26421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-hash 13310  df-edg 26137  df-uhgr 26150  df-nbgr 26422
This theorem is referenced by:  uvtx2vtx1edgb  26502
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