Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdgn0frgrav2 Structured version   Visualization version   GIF version

Theorem vdgn0frgrav2 26551
 Description: A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
Assertion
Ref Expression
vdgn0frgrav2 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))

Proof of Theorem vdgn0frgrav2
Dummy variables 𝑎 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26519 . . . . . 6 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
21anim1i 590 . . . . 5 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (𝑉 USGrph 𝐸𝑁𝑉))
32adantr 480 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸𝑁𝑉))
4 vdusgraval 26434 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
53, 4syl 17 . . 3 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
6 usgrafun 25878 . . . . . . . . 9 (𝑉 USGrph 𝐸 → Fun 𝐸)
7 funfn 5833 . . . . . . . . 9 (Fun 𝐸𝐸 Fn dom 𝐸)
86, 7sylib 207 . . . . . . . 8 (𝑉 USGrph 𝐸𝐸 Fn dom 𝐸)
91, 8syl 17 . . . . . . 7 (𝑉 FriendGrph 𝐸𝐸 Fn dom 𝐸)
10 3cyclfrgrarn 26540 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
11 preq1 4212 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1211eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
13 preq2 4213 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1413eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
1512, 143anbi13d 1393 . . . . . . . . . . . . . . 15 (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)))
16152rexbidv 3039 . . . . . . . . . . . . . 14 (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)))
1716rspcva 3280 . . . . . . . . . . . . 13 ((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))
18 fvelrnb 6153 . . . . . . . . . . . . . . . . . . . 20 (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏}))
19 prid1g 4239 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁𝑉𝑁 ∈ {𝑁, 𝑏})
20 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐸𝑥) = {𝑁, 𝑏} → (𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ {𝑁, 𝑏}))
2119, 20syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁𝑉 → ((𝐸𝑥) = {𝑁, 𝑏} → 𝑁 ∈ (𝐸𝑥)))
2221reximdv 2999 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
2322a1dd 48 . . . . . . . . . . . . . . . . . . . . 21 (𝑁𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2423com12 32 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑁, 𝑏} → (𝑁𝑉 → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2518, 24syl6bi 242 . . . . . . . . . . . . . . . . . . 19 (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑁𝑉 → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2625com12 32 . . . . . . . . . . . . . . . . . 18 ({𝑁, 𝑏} ∈ ran 𝐸 → (𝐸 Fn dom 𝐸 → (𝑁𝑉 → (𝑏𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2726com24 93 . . . . . . . . . . . . . . . . 17 ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑏𝑉 → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
28273ad2ant1 1075 . . . . . . . . . . . . . . . 16 (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑏𝑉 → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2928com12 32 . . . . . . . . . . . . . . 15 (𝑏𝑉 → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3029adantr 480 . . . . . . . . . . . . . 14 ((𝑏𝑉𝑐𝑉) → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3130rexlimivv 3018 . . . . . . . . . . . . 13 (∃𝑏𝑉𝑐𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
3217, 31syl 17 . . . . . . . . . . . 12 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
3332ex 449 . . . . . . . . . . 11 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3433pm2.43b 53 . . . . . . . . . 10 (∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
3510, 34syl 17 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
3635expcom 450 . . . . . . . 8 (1 < (#‘𝑉) → (𝑉 FriendGrph 𝐸 → (𝑁𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3736com14 94 . . . . . . 7 (𝐸 Fn dom 𝐸 → (𝑉 FriendGrph 𝐸 → (𝑁𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
389, 37mpcom 37 . . . . . 6 (𝑉 FriendGrph 𝐸 → (𝑁𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
3938imp31 447 . . . . 5 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
40 rexnal 2978 . . . . . 6 (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
41 notnotb 303 . . . . . . . 8 (𝑁 ∈ (𝐸𝑥) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑥))
4241bicomi 213 . . . . . . 7 (¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ (𝐸𝑥))
4342rexbii 3023 . . . . . 6 (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
4440, 43bitr3i 265 . . . . 5 (¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
4539, 44sylibr 223 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
46 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
47 dmexg 6989 . . . . . . . . . . 11 (𝐸 ∈ V → dom 𝐸 ∈ V)
4847adantl 481 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → dom 𝐸 ∈ V)
491, 46, 483syl 18 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → dom 𝐸 ∈ V)
5049adantr 480 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝑁𝑉) → dom 𝐸 ∈ V)
5150adantr 480 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ V)
52 rabexg 4739 . . . . . . 7 (dom 𝐸 ∈ V → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
53 hasheq0 13015 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅))
5451, 52, 533syl 18 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅))
55 rabeq0 3911 . . . . . 6 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = ∅ ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥))
5654, 55syl6bb 275 . . . . 5 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 0 ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥)))
5756necon3abid 2818 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 0 ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸𝑥)))
5845, 57mpbird 246 . . 3 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 0)
595, 58eqnetrd 2849 . 2 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0)
6059ex 449 1 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (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  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:  vdgfrgragt2  26554  frgraregord013  26645
 Copyright terms: Public domain W3C validator