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Theorem vdn1frgrav2 26552
 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.)
Assertion
Ref Expression
vdn1frgrav2 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))

Proof of Theorem vdn1frgrav2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26519 . . . . . . 7 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
213ad2ant1 1075 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝑉 USGrph 𝐸)
3 simp3 1056 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝑁𝑉)
4 simp2 1055 . . . . . 6 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → 𝐸 ∈ Fin)
52, 3, 43jca 1235 . . . . 5 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin))
65adantr 480 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin))
7 vdusgraval 26434 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
873adant3 1074 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉𝐸 ∈ Fin) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
96, 8syl 17 . . 3 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
10 3cyclfrgrarn2 26541 . . . . . 6 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
11103ad2antl1 1216 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
12 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1312eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
14 preq2 4213 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1514eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
1613, 153anbi13d 1393 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
1716anbi2d 736 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
18172rexbidv 3039 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
1918rspcva 3280 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
201adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑉 USGrph 𝐸)
21 simplr 788 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑁𝑉)
22 simplll 794 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑏𝑐)
23 3simpb 1052 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
2423ad3antlr 763 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
25 usgra2edg1 25912 . . . . . . . . . . . . . . . . 17 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
2620, 21, 22, 24, 25syl31anc 1321 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
27262a1d 26 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2827ex 449 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2928ex 449 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3029a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))))
3130rexlimivv 3018 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3219, 31syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3332ex 449 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))))
3433pm2.43a 52 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (𝐸 ∈ Fin → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3534com25 97 . . . . . . 7 (𝑁𝑉 → (𝐸 ∈ Fin → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3635com13 86 . . . . . 6 (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁𝑉 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
37363imp1 1272 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
3811, 37mpd 15 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
39 dmfi 8129 . . . . . . . . . 10 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
40393ad2ant2 1076 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → dom 𝐸 ∈ Fin)
4140adantr 480 . . . . . . . 8 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ Fin)
42 rabexg 4739 . . . . . . . 8 (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
4341, 42syl 17 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
44 hash1snb 13068 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
4543, 44syl 17 . . . . . 6 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
46 reusn 4206 . . . . . 6 (∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖})
4745, 46syl6bbr 277 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4847necon3abid 2818 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4938, 48mpbird 246 . . 3 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1)
509, 49eqnetrd 2849 . 2 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1)
5150ex 449 1 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   < clt 9953  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-vdgr 26421  df-frgra 26516 This theorem is referenced by: (None)
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