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Theorem nbgra0nb 25958
 Description: A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgra0nb (𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐸   𝑥,𝑁

Proof of Theorem nbgra0nb
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 nbusgra 25957 . . . 4 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
21adantr 480 . . 3 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
3 neleq2 2889 . . . . . . . . . 10 (𝑥 = {𝑁, 𝑛} → (𝑁𝑥𝑁 ∉ {𝑁, 𝑛}))
43rspcv 3278 . . . . . . . . 9 ({𝑁, 𝑛} ∈ ran 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥𝑁 ∉ {𝑁, 𝑛}))
5 df-nel 2783 . . . . . . . . . 10 (𝑁 ∉ {𝑁, 𝑛} ↔ ¬ 𝑁 ∈ {𝑁, 𝑛})
6 elprg 4144 . . . . . . . . . . . . . 14 (𝑁𝑉 → (𝑁 ∈ {𝑁, 𝑛} ↔ (𝑁 = 𝑁𝑁 = 𝑛)))
76notbid 307 . . . . . . . . . . . . 13 (𝑁𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} ↔ ¬ (𝑁 = 𝑁𝑁 = 𝑛)))
8 ioran 510 . . . . . . . . . . . . 13 (¬ (𝑁 = 𝑁𝑁 = 𝑛) ↔ (¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛))
97, 8syl6bb 275 . . . . . . . . . . . 12 (𝑁𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} ↔ (¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛)))
10 eqid 2610 . . . . . . . . . . . . . 14 𝑁 = 𝑁
1110pm2.24i 145 . . . . . . . . . . . . 13 𝑁 = 𝑁 → (¬ 𝑁 = 𝑛 → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
1211imp 444 . . . . . . . . . . . 12 ((¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛) → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
139, 12syl6bi 242 . . . . . . . . . . 11 (𝑁𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
14 usgraedgrnv 25906 . . . . . . . . . . . . . . . . 17 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → (𝑁𝑉𝑛𝑉))
1514simpld 474 . . . . . . . . . . . . . . . 16 ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → 𝑁𝑉)
1615ex 449 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸𝑁𝑉))
1716con3d 147 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → (¬ 𝑁𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
1817a1i 11 . . . . . . . . . . . . 13 (𝑛𝑉 → (𝑉 USGrph 𝐸 → (¬ 𝑁𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
1918com13 86 . . . . . . . . . . . 12 𝑁𝑉 → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
2019a1d 25 . . . . . . . . . . 11 𝑁𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
2113, 20pm2.61i 175 . . . . . . . . . 10 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
225, 21sylbi 206 . . . . . . . . 9 (𝑁 ∉ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
234, 22syl6 34 . . . . . . . 8 ({𝑁, 𝑛} ∈ ran 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥 → (𝑉 USGrph 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
2423com13 86 . . . . . . 7 (𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥 → ({𝑁, 𝑛} ∈ ran 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
2524imp 444 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → ({𝑁, 𝑛} ∈ ran 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
26 ax-1 6 . . . . . 6 (¬ {𝑁, 𝑛} ∈ ran 𝐸 → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
2725, 26pm2.61d1 170 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → (𝑛𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
2827ralrimiv 2948 . . . 4 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
29 rabeq0 3911 . . . 4 ({𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅ ↔ ∀𝑛𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
3028, 29sylibr 223 . . 3 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅)
312, 30eqtrd 2644 . 2 ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁𝑥) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
3231ex 449 1 (𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁𝑥 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900  ∅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: (None)
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