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Theorem vdgn1frgrv2 30324
Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)
Hypothesis
Ref Expression
vdn1frgrv2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
vdgn1frgrv2 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrv2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrusgr 30289 . . . . . 6 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
21anim1i 615 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
32adantr 480 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
4 vdn1frgrv2.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 eqid 2734 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2734 . . . . 5 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7 eqid 2734 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
84, 5, 6, 7vtxdusgrval 29519 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
93, 8syl 17 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
10 eqid 2734 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
114, 103cyclfrgrrn2 30315 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
1211adantlr 715 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
13 preq1 4737 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1413eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15 preq2 4738 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1615eleq1d 2823 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
1714, 163anbi13d 1437 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)) ↔ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
1817anbi2d 630 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
19182rexbidv 3219 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
2019rspcva 3619 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
211adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ USGraph)
22 simplll 775 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑏𝑐)
23 3simpb 1148 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
2423ad3antlr 731 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
255, 10usgr2edg1 29243 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USGraph ∧ 𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2621, 22, 24, 25syl21anc 838 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2726a1d 25 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2827ex 412 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))
2928ex 412 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3029a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3130rexlimivv 3198 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3220, 31syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3332ex 412 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3433pm2.43a 54 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3534com24 95 . . . . . . 7 (𝑁𝑉 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3635com3r 87 . . . . . 6 (𝐺 ∈ FriendGraph → (𝑁𝑉 → (1 < (♯‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3736imp31 417 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
3812, 37mpd 15 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
39 fvex 6919 . . . . . . . . 9 (iEdg‘𝐺) ∈ V
4039dmex 7931 . . . . . . . 8 dom (iEdg‘𝐺) ∈ V
4140a1i 11 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42 rabexg 5342 . . . . . . 7 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43 hash1snb 14454 . . . . . . 7 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
4441, 42, 433syl 18 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
45 reusn 4731 . . . . . 6 (∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖})
4644, 45bitr4di 289 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4746necon3abid 2974 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4838, 47mpbird 257 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1)
499, 48eqnetrd 3005 . 2 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (♯‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1)
5049ex 412 1 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (♯‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  ∃!wreu 3375  {crab 3432  Vcvv 3477  {csn 4630  {cpr 4632   class class class wbr 5147  dom cdm 5688  cfv 6562  1c1 11153   < clt 11292  chash 14365  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  USGraphcusgr 29180  VtxDegcvtxdg 29497   FriendGraph cfrgr 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-xadd 13152  df-fz 13544  df-hash 14366  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-umgr 29114  df-usgr 29182  df-vtxdg 29498  df-frgr 30287
This theorem is referenced by:  vdgn1frgrv3  30325  vdgfrgrgt2  30326
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