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Theorem nbusgra 25957
 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 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 nbgraop 25952 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
32ex 449 . . . 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 2783 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
7 nbgranv0 25956 . . . . . 6 (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
86, 7sylbir 224 . . . . 5 𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
98adantr 480 . . . 4 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
10 usgraedgrnv 25906 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → (𝑁𝑉𝑛𝑉))
11 notnot 135 . . . . . . . . . . . . 13 (𝑁𝑉 → ¬ ¬ 𝑁𝑉)
1211adantr 480 . . . . . . . . . . . 12 ((𝑁𝑉𝑛𝑉) → ¬ ¬ 𝑁𝑉)
1312intnand 953 . . . . . . . . . . 11 ((𝑁𝑉𝑛𝑉) → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉))
1410, 13syl 17 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉))
1514ex 449 . . . . . . . . 9 (𝑉 USGrph 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 → ¬ (𝑛𝑉 ∧ ¬ 𝑁𝑉)))
1615con2d 128 . . . . . . . 8 (𝑉 USGrph 𝐸 → ((𝑛𝑉 ∧ ¬ 𝑁𝑉) → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
1716expcomd 453 . . . . . . 7 (𝑉 USGrph 𝐸 → (¬ 𝑁𝑉 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
1817impcom 445 . . . . . 6 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
1918ralrimiv 2948 . . . . 5 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
20 rabeq0 3911 . . . . 5 ({𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅ ↔ ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
2119, 20sylibr 223 . . . 4 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅)
229, 21eqtr4d 2647 . . 3 ((¬ 𝑁𝑉𝑉 USGrph 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
2322ex 449 . 2 𝑁𝑉 → (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
245, 23pm2.61i 175 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 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949 This theorem is referenced by:  nbgra0nb  25958  nbgraeledg  25959  nbgra0edg  25961  nbgrassvt  25962  nbgranself  25963  nb3graprlem1  25980  nbcusgra  25992  cusgrasizeindslem2  26003  uvtxnbgra  26021  uvtxnb  26025  frisusgranb  26524
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