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Theorem nbgranself 25726
Description: A vertex in a graph (without loops!) is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgranself (𝑉 USGrph 𝐸 → ∀𝑣𝑉 𝑣 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))
Distinct variable groups:   𝑣,𝑉   𝑣,𝐸

Proof of Theorem nbgranself
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 usgraedgrn 25673 . . . . . . . . 9 ((𝑉 USGrph 𝐸 ∧ {𝑣, 𝑣} ∈ ran 𝐸) → 𝑣𝑣)
2 df-ne 2778 . . . . . . . . . 10 (𝑣𝑣 ↔ ¬ 𝑣 = 𝑣)
3 equid 1925 . . . . . . . . . . 11 𝑣 = 𝑣
43pm2.24i 144 . . . . . . . . . 10 𝑣 = 𝑣 → ¬ 𝑣𝑉)
52, 4sylbi 205 . . . . . . . . 9 (𝑣𝑣 → ¬ 𝑣𝑉)
61, 5syl 17 . . . . . . . 8 ((𝑉 USGrph 𝐸 ∧ {𝑣, 𝑣} ∈ ran 𝐸) → ¬ 𝑣𝑉)
76ex 448 . . . . . . 7 (𝑉 USGrph 𝐸 → ({𝑣, 𝑣} ∈ ran 𝐸 → ¬ 𝑣𝑉))
87con2d 127 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑣𝑉 → ¬ {𝑣, 𝑣} ∈ ran 𝐸))
98imp 443 . . . . 5 ((𝑉 USGrph 𝐸𝑣𝑉) → ¬ {𝑣, 𝑣} ∈ ran 𝐸)
10 preq2 4209 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
1110eleq1d 2668 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ∈ ran 𝐸 ↔ {𝑣, 𝑣} ∈ ran 𝐸))
1211elrab3 3328 . . . . . 6 (𝑣𝑉 → (𝑣 ∈ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸} ↔ {𝑣, 𝑣} ∈ ran 𝐸))
1312adantl 480 . . . . 5 ((𝑉 USGrph 𝐸𝑣𝑉) → (𝑣 ∈ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸} ↔ {𝑣, 𝑣} ∈ ran 𝐸))
149, 13mtbird 313 . . . 4 ((𝑉 USGrph 𝐸𝑣𝑉) → ¬ 𝑣 ∈ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
15 df-nel 2779 . . . 4 (𝑣 ∉ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸} ↔ ¬ 𝑣 ∈ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
1614, 15sylibr 222 . . 3 ((𝑉 USGrph 𝐸𝑣𝑉) → 𝑣 ∉ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
17 eqidd 2607 . . . 4 ((𝑉 USGrph 𝐸𝑣𝑉) → 𝑣 = 𝑣)
18 nbusgra 25720 . . . . 5 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
1918adantr 479 . . . 4 ((𝑉 USGrph 𝐸𝑣𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) = {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
2017, 19neleq12d 2883 . . 3 ((𝑉 USGrph 𝐸𝑣𝑉) → (𝑣 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) ↔ 𝑣 ∉ {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}))
2116, 20mpbird 245 . 2 ((𝑉 USGrph 𝐸𝑣𝑉) → 𝑣 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))
2221ralrimiva 2945 1 (𝑉 USGrph 𝐸 → ∀𝑣𝑉 𝑣 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2776  wnel 2777  wral 2892  {crab 2896  {cpr 4123  cop 4127   class class class wbr 4574  ran crn 5026  (class class class)co 6524   USGrph cusg 25622   Neighbors cnbgra 25709
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 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-hash 12932  df-usgra 25625  df-nbgra 25712
This theorem is referenced by:  nbgrassovt  25727
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