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Theorem nbhashuvtx 26443
 Description: If the number of the neighbors of a vertex in a graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
Assertion
Ref Expression
nbhashuvtx ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑁 ∈ (𝑉 UnivVertex 𝐸)))

Proof of Theorem nbhashuvtx
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nbgrassvwo 25966 . . . . . . . 8 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))
21sseld 3567 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
323ad2ant1 1075 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
43adantr 480 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → (𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) → 𝑣 ∈ (𝑉 ∖ {𝑁})))
5 eldifsn 4260 . . . . . 6 (𝑣 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑣𝑉𝑣𝑁))
6 nbhashuvtx1 26442 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑣𝑉𝑣𝑁) → 𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
76imp 444 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → ((𝑣𝑉𝑣𝑁) → 𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
85, 7syl5bi 231 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)))
94, 8impbid 201 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → (𝑣 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ 𝑣 ∈ (𝑉 ∖ {𝑁})))
109eqrdv 2608 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
11 uvtxnb 26025 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
12113adant2 1073 . . . 4 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
1312adantr 480 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))
1410, 13mpbird 246 . 2 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)) → 𝑁 ∈ (𝑉 UnivVertex 𝐸))
1514ex 449 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑁 ∈ (𝑉 UnivVertex 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  {csn 4125  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   − cmin 10145  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946   UnivVertex cuvtx 25948 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-2o 7448  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-nbgra 25949  df-uvtx 25951  df-vdgr 26421 This theorem is referenced by:  uvtxhashnb  26444
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