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Theorem nbuhgr2vtx1edgb 26135
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.)
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)
2 fvex 6158 . . . . 5 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2694 . . . 4 𝑉 ∈ V
4 hash2prb 13192 . . . 4 (𝑉 ∈ V → ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
53, 4ax-mp 5 . . 3 ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}))
6 simpr 477 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑎𝑉𝑏𝑉))
76ancomd 467 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑏𝑉𝑎𝑉))
87ad2antrr 761 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏𝑉𝑎𝑉))
9 id 22 . . . . . . . . . . . . 13 (𝑎𝑏𝑎𝑏)
109necomd 2845 . . . . . . . . . . . 12 (𝑎𝑏𝑏𝑎)
1110adantr 481 . . . . . . . . . . 11 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → 𝑏𝑎)
1211ad2antlr 762 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏𝑎)
13 prcom 4237 . . . . . . . . . . . . . 14 {𝑎, 𝑏} = {𝑏, 𝑎}
1413eleq1i 2689 . . . . . . . . . . . . 13 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
1514biimpi 206 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ∈ 𝐸)
16 sseq2 3606 . . . . . . . . . . . . 13 (𝑒 = {𝑏, 𝑎} → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1716adantl 482 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑏, 𝑎}) → ({𝑎, 𝑏} ⊆ 𝑒 ↔ {𝑎, 𝑏} ⊆ {𝑏, 𝑎}))
1813eqimssi 3638 . . . . . . . . . . . . 13 {𝑎, 𝑏} ⊆ {𝑏, 𝑎}
1918a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ⊆ {𝑏, 𝑎})
2015, 17, 19rspcedvd 3302 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
2120adantl 482 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)
22 nbgr2vtx1edg.e . . . . . . . . . . . 12 𝐸 = (Edg‘𝐺)
231, 22nbgrel 26125 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)))
2423ad3antrrr 765 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ ((𝑏𝑉𝑎𝑉) ∧ 𝑏𝑎 ∧ ∃𝑒𝐸 {𝑎, 𝑏} ⊆ 𝑒)))
258, 12, 21, 24mpbir3and 1243 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
266ad2antrr 761 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎𝑉𝑏𝑉))
27 simplrl 799 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎𝑏)
28 id 22 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑎, 𝑏} ∈ 𝐸)
29 sseq2 3606 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
3029adantl 482 . . . . . . . . . . . 12 (({𝑎, 𝑏} ∈ 𝐸𝑒 = {𝑎, 𝑏}) → ({𝑏, 𝑎} ⊆ 𝑒 ↔ {𝑏, 𝑎} ⊆ {𝑎, 𝑏}))
31 prcom 4237 . . . . . . . . . . . . . 14 {𝑏, 𝑎} = {𝑎, 𝑏}
3231eqimssi 3638 . . . . . . . . . . . . 13 {𝑏, 𝑎} ⊆ {𝑎, 𝑏}
3332a1i 11 . . . . . . . . . . . 12 ({𝑎, 𝑏} ∈ 𝐸 → {𝑏, 𝑎} ⊆ {𝑎, 𝑏})
3428, 30, 33rspcedvd 3302 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
3534adantl 482 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)
361, 22nbgrel 26125 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
3736ad3antrrr 765 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏 ∧ ∃𝑒𝐸 {𝑏, 𝑎} ⊆ 𝑒)))
3826, 27, 35, 37mpbir3and 1243 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
3925, 38jca 554 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) ∧ {𝑎, 𝑏} ∈ 𝐸) → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
4039ex 450 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
411, 22nbuhgr2vtx1edgblem 26134 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸)
42413exp 1261 . . . . . . . . . . 11 (𝐺 ∈ UHGraph → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4342adantr 481 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → (𝑉 = {𝑎, 𝑏} → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4443adantld 483 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
4544imp 445 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑎 ∈ (𝐺 NeighbVtx 𝑏) → {𝑎, 𝑏} ∈ 𝐸))
4645adantld 483 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ((𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸))
4740, 46impbid 202 . . . . . 6 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
48 eleq1 2686 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
4948adantl 482 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
50 id 22 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → 𝑉 = {𝑎, 𝑏})
51 difeq1 3699 . . . . . . . . . . 11 (𝑉 = {𝑎, 𝑏} → (𝑉 ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑣}))
5251raleqdv 3133 . . . . . . . . . 10 (𝑉 = {𝑎, 𝑏} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5350, 52raleqbidv 3141 . . . . . . . . 9 (𝑉 = {𝑎, 𝑏} → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
54 vex 3189 . . . . . . . . . . 11 𝑎 ∈ V
55 vex 3189 . . . . . . . . . . 11 𝑏 ∈ V
56 sneq 4158 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → {𝑣} = {𝑎})
5756difeq2d 3706 . . . . . . . . . . . 12 (𝑣 = 𝑎 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑎}))
58 oveq2 6612 . . . . . . . . . . . . 13 (𝑣 = 𝑎 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑎))
5958eleq2d 2684 . . . . . . . . . . . 12 (𝑣 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
6057, 59raleqbidv 3141 . . . . . . . . . . 11 (𝑣 = 𝑎 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
61 sneq 4158 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → {𝑣} = {𝑏})
6261difeq2d 3706 . . . . . . . . . . . 12 (𝑣 = 𝑏 → ({𝑎, 𝑏} ∖ {𝑣}) = ({𝑎, 𝑏} ∖ {𝑏}))
63 oveq2 6612 . . . . . . . . . . . . 13 (𝑣 = 𝑏 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑏))
6463eleq2d 2684 . . . . . . . . . . . 12 (𝑣 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6562, 64raleqbidv 3141 . . . . . . . . . . 11 (𝑣 = 𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
6654, 55, 60, 65ralpr 4209 . . . . . . . . . 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)))
67 difprsn1 4299 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑎}) = {𝑏})
6867raleqdv 3133 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ ∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎)))
69 eleq1 2686 . . . . . . . . . . . . 13 (𝑛 = 𝑏 → (𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
7055, 69ralsn 4193 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑏}𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
7168, 70syl6bb 276 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑎})𝑛 ∈ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)))
72 difprsn2 4300 . . . . . . . . . . . . 13 (𝑎𝑏 → ({𝑎, 𝑏} ∖ {𝑏}) = {𝑎})
7372raleqdv 3133 . . . . . . . . . . . 12 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ ∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏)))
74 eleq1 2686 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → (𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7554, 74ralsn 4193 . . . . . . . . . . . 12 (∀𝑛 ∈ {𝑎}𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))
7673, 75syl6bb 276 . . . . . . . . . . 11 (𝑎𝑏 → (∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑏})𝑛 ∈ (𝐺 NeighbVtx 𝑏) ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))
7771, 76anbi12d 746 . . . . . . . . . 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))))
7866, 77syl5bb 272 . . . . . . . . 9 (𝑎𝑏 → (∀𝑣 ∈ {𝑎, 𝑏}∀𝑛 ∈ ({𝑎, 𝑏} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
7953, 78sylan9bbr 736 . . . . . . . 8 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏))))
8049, 79bibi12d 335 . . . . . . 7 ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → ((𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)))))
8180adantl 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)))))
8247, 81mpbird 247 . . . . 5 (((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑎𝑏𝑉 = {𝑎, 𝑏})) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8382ex 450 . . . 4 ((𝐺 ∈ UHGraph ∧ (𝑎𝑉𝑏𝑉)) → ((𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8483rexlimdvva 3031 . . 3 (𝐺 ∈ UHGraph → (∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏}) → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
855, 84syl5bi 232 . 2 (𝐺 ∈ UHGraph → ((#‘𝑉) = 2 → (𝑉𝐸 ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))))
8685imp 445 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  {csn 4148  {cpr 4150  cfv 5847  (class class class)co 6604  2c2 11014  #chash 13057  Vtxcvtx 25774  Edgcedg 25839   UHGraph cuhgr 25847   NeighbVtx cnbgr 26111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-edg 25840  df-uhgr 25849  df-nbgr 26115
This theorem is referenced by:  uvtx2vtx1edgb  26187
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