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Theorem vdn0frgrav2 26550
 Description: A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
Assertion
Ref Expression
vdn0frgrav2 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))

Proof of Theorem vdn0frgrav2
Dummy variables 𝑎 𝑏 𝑐 𝑥 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 usgrafun 25878 . . . . . . . . 9 (𝑉 USGrph 𝐸 → Fun 𝐸)
11 funfn 5833 . . . . . . . . 9 (Fun 𝐸𝐸 Fn dom 𝐸)
1210, 11sylib 207 . . . . . . . 8 (𝑉 USGrph 𝐸𝐸 Fn dom 𝐸)
131, 12syl 17 . . . . . . 7 (𝑉 FriendGrph 𝐸𝐸 Fn dom 𝐸)
14 3cyclfrgrarn 26540 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
15 preq1 4212 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1615eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
17 preq2 4213 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1817eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
1916, 183anbi13d 1393 . . . . . . . . . . . . . . . 16 (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)))
20192rexbidv 3039 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏𝑉𝑐𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
2120rspcva 3280 . . . . . . . . . . . . . 14 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ∃𝑏𝑉𝑐𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
22 fvelrnb 6153 . . . . . . . . . . . . . . . . . . . . 21 (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏}))
23 prid1g 4239 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁𝑉𝑁 ∈ {𝑁, 𝑏})
24 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐸𝑥) = {𝑁, 𝑏} → (𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ {𝑁, 𝑏}))
2523, 24syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁𝑉 → ((𝐸𝑥) = {𝑁, 𝑏} → 𝑁 ∈ (𝐸𝑥)))
2625reximdv 2999 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
2726a1dd 48 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2827a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸 ∈ Fin → (𝑁𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2928com13 86 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → (𝑁𝑉 → (𝐸 ∈ Fin → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3022, 29syl6bi 242 . . . . . . . . . . . . . . . . . . . 20 (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑁𝑉 → (𝐸 ∈ Fin → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3130com12 32 . . . . . . . . . . . . . . . . . . 19 ({𝑁, 𝑏} ∈ ran 𝐸 → (𝐸 Fn dom 𝐸 → (𝑁𝑉 → (𝐸 ∈ Fin → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3231com25 97 . . . . . . . . . . . . . . . . . 18 ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑏𝑉 → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
33323ad2ant1 1075 . . . . . . . . . . . . . . . . 17 (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑏𝑉 → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3433com12 32 . . . . . . . . . . . . . . . 16 (𝑏𝑉 → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3534adantr 480 . . . . . . . . . . . . . . 15 ((𝑏𝑉𝑐𝑉) → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3635rexlimivv 3018 . . . . . . . . . . . . . 14 (∃𝑏𝑉𝑐𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3721, 36syl 17 . . . . . . . . . . . . 13 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3837ex 449 . . . . . . . . . . . 12 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3938pm2.43a 52 . . . . . . . . . . 11 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
4039com3l 87 . . . . . . . . . 10 (∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝐸 ∈ Fin → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
4114, 40syl 17 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → (𝐸 ∈ Fin → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
4241expcom 450 . . . . . . . 8 (1 < (#‘𝑉) → (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
4342com15 99 . . . . . . 7 (𝐸 Fn dom 𝐸 → (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
4413, 43mpcom 37 . . . . . 6 (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
45443imp1 1272 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
46 rexnal 2978 . . . . . 6 (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
47 notnotb 303 . . . . . . . 8 (𝑁 ∈ (𝐸𝑥) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑥))
4847bicomi 213 . . . . . . 7 (¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ (𝐸𝑥))
4948rexbii 3023 . . . . . 6 (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
5046, 49bitr3i 265 . . . . 5 (¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
5145, 50sylibr 223 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
52 dmfi 8129 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
53523ad2ant2 1076 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → dom 𝐸 ∈ Fin)
5453adantr 480 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ Fin)
55 rabexg 4739 . . . . . . 7 (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
56 hasheq0 13015 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅))
5754, 55, 563syl 18 . . . . . 6 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅))
58 rabeq0 3911 . . . . . 6 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅ ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
5957, 58syl6bb 275 . . . . 5 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥)))
6059necon3abid 2818 . . . 4 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 0 ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥)))
6151, 60mpbird 246 . . 3 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 0)
629, 61eqnetrd 2849 . 2 (((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0)
6362ex 449 1 ((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  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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