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

Theorem nbusgra 25750
Description: The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Proof shortened by Alexander van der Vekens, 25-Jan-2018.)
Assertion
Ref Expression
nbusgra (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
Distinct variable groups:   𝑛,𝑉   𝑛,𝐸   𝑛,𝑁

Proof of Theorem nbusgra
StepHypRef Expression
1 usgrav 25660 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 nbgraop 25745 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
32ex 448 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
41, 3syl 17 . . 3 (𝑉 USGrph 𝐸 → (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
54com12 32 . 2 (𝑁𝑉 → (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
6 df-nel 2782 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
7 nbgranv0 25749 . . . . . 6 (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
86, 7sylbir 223 . . . . 5 𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
98adantr 479 . . . 4 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
10 usgraedgrnv 25699 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → (𝑁𝑉𝑛𝑉))
11 notnot 134 . . . . . . . . . . . . 13 (𝑁𝑉 → ¬ ¬ 𝑁𝑉)
1211adantr 479 . . . . . . . . . . . 12 ((𝑁𝑉𝑛𝑉) → ¬ ¬ 𝑁𝑉)
1312intnand 952 . . . . . . . . . . 11 ((𝑁𝑉𝑛𝑉) → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉))
1410, 13syl 17 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉))
1514ex 448 . . . . . . . . 9 (𝑉 USGrph 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉)))
1615con2d 127 . . . . . . . 8 (𝑉 USGrph 𝐸 → ((𝑛𝑉 ∧ ¬ 𝑁𝑉) → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
1716expcomd 452 . . . . . . 7 (𝑉 USGrph 𝐸 → (¬ 𝑁𝑉 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
1817impcom 444 . . . . . 6 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
1918ralrimiv 2947 . . . . 5 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
20 rabeq0 3910 . . . . 5 ({𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅ ↔ ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
2119, 20sylibr 222 . . . 4 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅)
229, 21eqtr4d 2646 . . 3 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
2322ex 448 . 2 𝑁𝑉 → (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
245, 23pm2.61i 174 1 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wnel 2780  wral 2895  {crab 2899  Vcvv 3172  c0 3873  {cpr 4126  cop 4130   class class class wbr 4577  ran crn 5028  (class class class)co 6526   USGrph cusg 25652   Neighbors cnbgra 25739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-n0 11142  df-z 11213  df-uz 11522  df-fz 12155  df-hash 12937  df-usgra 25655  df-nbgra 25742
This theorem is referenced by:  nbgra0nb  25751  nbgraeledg  25752  nbgra0edg  25754  nbgrassvt  25755  nbgranself  25756  nb3graprlem1  25773  nbcusgra  25785  cusgrasizeindslem2  25796  uvtxnbgra  25814  uvtxnb  25818  frisusgranb  26317
  Copyright terms: Public domain W3C validator