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Theorem nbuhgr2vtx1edgb 27050
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 6680 . . . 4 𝑉 ∈ V
3 hash2prb 13823 . . . 4 (𝑉 ∈ V → ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
42, 3ax-mp 5 . . 3 ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}))
5 simpr 485 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑉𝑏𝑉))
65ancomd 462 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑉𝑎𝑉))
76ad2antrr 722 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏𝑉𝑎𝑉))
8 id 22 . . . . . . . . . . . . 13 (𝑎𝑏𝑎𝑏)
98necomd 3075 . . . . . . . . . . . 12 (𝑎𝑏𝑏𝑎)
109adantr 481 . . . . . . . . . . 11 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑏𝑎)
1110ad2antlr 723 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏𝑎)
12 prcom 4666 . . . . . . . . . . . . . 14 {𝑎, 𝑏} = {𝑏, 𝑎}
1312eleq1i 2907 . . . . . . . . . . . . 13 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
1413biimpi 217 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
15 sseq2 3996 . . . . . . . . . . . . 13 (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1615adantl 482 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1712eqimssi 4028 . . . . . . . . . . . . 13 {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
1817a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
1914, 16, 18rspcedvd 3629 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
2019adantl 482 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
21 nbgr2vtx1edg.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
221, 21nbgrel 27038 . . . . . . . . . 10 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
237, 11, 20, 22syl3anbrc 1337 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
245ad2antrr 722 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎𝑉𝑏𝑉))
25 simplrl 773 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎𝑏)
26 id 22 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
27 sseq2 3996 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
2827adantl 482 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
29 prcom 4666 . . . . . . . . . . . . . 14 {𝑏, 𝑎} = {𝑎, 𝑏}
3029eqimssi 4028 . . . . . . . . . . . . 13 {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
3130a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
3226, 28, 31rspcedvd 3629 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
3332adantl 482 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
341, 21nbgrel 27038 . . . . . . . . . 10 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
3524, 25, 33, 34syl3anbrc 1337 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
3623, 35jca 512 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
3736ex 413 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
381, 21nbuhgr2vtx1edgblem 27049 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4140adantld 491 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4241imp 407 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
4342adantld 491 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ((𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸))
4437, 43impbid 213 . . . . . 6 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
45 eleq1 2904 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
4645adantl 482 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
47 id 22 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
48 difeq1 4095 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
4948raleqdv 3420 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5047, 49raleqbidv 3406 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51 vex 3502 . . . . . . . . . . 11 𝑎 ∈ V
52 vex 3502 . . . . . . . . . . 11 𝑏 ∈ V
53 sneq 4573 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → {𝑣} = {𝑎})
5453difeq2d 4102 . . . . . . . . . . . 12 (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
55 oveq2 7159 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
5655eleq2d 2902 . . . . . . . . . . . 12 (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
5754, 56raleqbidv 3406 . . . . . . . . . . 11 (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
58 sneq 4573 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑣} = {𝑏})
5958difeq2d 4102 . . . . . . . . . . . 12 (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
60 oveq2 7159 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
6160eleq2d 2902 . . . . . . . . . . . 12 (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6259, 61raleqbidv 3406 . . . . . . . . . . 11 (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6351, 52, 57, 62ralpr 4634 . . . . . . . . . 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 4731 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
6564raleqdv 3420 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
66 eleq1 2904 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
6752, 66ralsn 4617 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
6865, 67syl6bb 288 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
69 difprsn2 4732 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
7069raleqdv 3420 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
71 eleq1 2904 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7251, 71ralsn 4617 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
7370, 72syl6bb 288 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7468, 73anbi12d 630 . . . . . . . . . 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 284 . . . . . . . . 9 (𝑎𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7650, 75sylan9bbr 511 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7746, 76bibi12d 347 . . . . . . 7 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → ((𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
7877adantl 482 . . . . . 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 258 . . . . 5 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8079ex 413 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8180rexlimdvva 3298 . . 3 (𝐺 ∈ UHGraph → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
824, 81syl5bi 243 . 2 (𝐺 ∈ UHGraph → ((♯‘𝑉) = 2 → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8382imp 407 1 ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3020  wral 3142  wrex 3143  Vcvv 3499  cdif 3936  wss 3939  {csn 4563  {cpr 4565  cfv 6351  (class class class)co 7151  2c2 11684  chash 13683  Vtxcvtx 26697  Edgcedg 26748  UHGraphcuhgr 26757   NeighbVtx cnbgr 27030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-edg 26749  df-uhgr 26759  df-nbgr 27031
This theorem is referenced by:  uvtx2vtx1edgb  27097
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