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Theorem nbuhgr2vtx1edgb 26527
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 6389 . . . 4 𝑉 ∈ V
3 hash2prb 13455 . . . 4 (𝑉 ∈ V → ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
42, 3ax-mp 5 . . 3 ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}))
5 simpr 477 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑉𝑏𝑉))
65ancomd 453 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑉𝑎𝑉))
76ad2antrr 717 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏𝑉𝑎𝑉))
8 id 22 . . . . . . . . . . . . 13 (𝑎𝑏𝑎𝑏)
98necomd 2992 . . . . . . . . . . . 12 (𝑎𝑏𝑏𝑎)
109adantr 472 . . . . . . . . . . 11 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑏𝑎)
1110ad2antlr 718 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏𝑎)
12 prcom 4422 . . . . . . . . . . . . . 14 {𝑎, 𝑏} = {𝑏, 𝑎}
1312eleq1i 2835 . . . . . . . . . . . . 13 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
1413biimpi 207 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
15 sseq2 3787 . . . . . . . . . . . . 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 3819 . . . . . . . . . . . . 13 {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
1817a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
1914, 16, 18rspcedvd 3468 . . . . . . . . . . 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 26512 . . . . . . . . . 10 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒))
237, 11, 20, 22syl3anbrc 1443 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
245ad2antrr 717 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎𝑉𝑏𝑉))
25 simplrl 795 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎𝑏)
26 id 22 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
27 sseq2 3787 . . . . . . . . . . . . 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 4422 . . . . . . . . . . . . . 14 {𝑏, 𝑎} = {𝑎, 𝑏}
3029eqimssi 3819 . . . . . . . . . . . . 13 {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
3130a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
3226, 28, 31rspcedvd 3468 . . . . . . . . . . 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 26512 . . . . . . . . . 10 (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒))
3524, 25, 33, 34syl3anbrc 1443 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
3623, 35jca 507 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
3736ex 401 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
381, 21nbuhgr2vtx1edgblem 26526 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
39383exp 1148 . . . . . . . . . . 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 395 . . . . . . . 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 203 . . . . . 6 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
45 eleq1 2832 . . . . . . . . 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 3883 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
4948raleqdv 3292 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5047, 49raleqbidv 3300 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
51 vex 3353 . . . . . . . . . . 11 𝑎 ∈ V
52 vex 3353 . . . . . . . . . . 11 𝑏 ∈ V
53 sneq 4344 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → {𝑣} = {𝑎})
5453difeq2d 3890 . . . . . . . . . . . 12 (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
55 oveq2 6850 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
5655eleq2d 2830 . . . . . . . . . . . 12 (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
5754, 56raleqbidv 3300 . . . . . . . . . . 11 (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
58 sneq 4344 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑣} = {𝑏})
5958difeq2d 3890 . . . . . . . . . . . 12 (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
60 oveq2 6850 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
6160eleq2d 2830 . . . . . . . . . . . 12 (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6259, 61raleqbidv 3300 . . . . . . . . . . 11 (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6351, 52, 57, 62ralpr 4394 . . . . . . . . . 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 4485 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
6564raleqdv 3292 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
66 eleq1 2832 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
6752, 66ralsn 4379 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
6865, 67syl6bb 278 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
69 difprsn2 4486 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
7069raleqdv 3292 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
71 eleq1 2832 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7251, 71ralsn 4379 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
7370, 72syl6bb 278 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7468, 73anbi12d 624 . . . . . . . . . 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 274 . . . . . . . . 9 (𝑎𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7650, 75sylan9bbr 506 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7746, 76bibi12d 336 . . . . . . 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 248 . . . . 5 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8079ex 401 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8180rexlimdvva 3185 . . 3 (𝐺 ∈ UHGraph → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
824, 81syl5bi 233 . 2 (𝐺 ∈ UHGraph → ((♯‘𝑉) = 2 → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8382imp 395 1 ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  wss 3732  {csn 4334  {cpr 4336  cfv 6068  (class class class)co 6842  2c2 11327  chash 13321  Vtxcvtx 26165  Edgcedg 26216  UHGraphcuhgr 26228   NeighbVtx cnbgr 26503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-edg 26217  df-uhgr 26230  df-nbgr 26504
This theorem is referenced by:  uvtx2vtx1edgb  26585
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