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 𝑁 ∉ 𝑥 → (⟨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.

Step	Hyp	Ref	Expression
1		nbusgra 25957	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
2	1	adantr 480	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
3		neleq2 2889	. . . . . . . . . 10 ⊢ (𝑥 = {𝑁, 𝑛} → (𝑁 ∉ 𝑥 ↔ 𝑁 ∉ {𝑁, 𝑛}))
4	3	rspcv 3278	. . . . . . . . 9 ⊢ ({𝑁, 𝑛} ∈ ran 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥 → 𝑁 ∉ {𝑁, 𝑛}))
5		df-nel 2783	. . . . . . . . . 10 ⊢ (𝑁 ∉ {𝑁, 𝑛} ↔ ¬ 𝑁 ∈ {𝑁, 𝑛})
6		elprg 4144	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ {𝑁, 𝑛} ↔ (𝑁 = 𝑁 ∨ 𝑁 = 𝑛)))
7	6	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ 𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} ↔ ¬ (𝑁 = 𝑁 ∨ 𝑁 = 𝑛)))
8		ioran 510	. . . . . . . . . . . . 13 ⊢ (¬ (𝑁 = 𝑁 ∨ 𝑁 = 𝑛) ↔ (¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛))
9	7, 8	syl6bb 275	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} ↔ (¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛)))
10		eqid 2610	. . . . . . . . . . . . . 14 ⊢ 𝑁 = 𝑁
11	10	pm2.24i 145	. . . . . . . . . . . . 13 ⊢ (¬ 𝑁 = 𝑁 → (¬ 𝑁 = 𝑛 → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
12	11	imp 444	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 𝑁 ∧ ¬ 𝑁 = 𝑛) → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
13	9, 12	syl6bi 242	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
14		usgraedgrnv 25906	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
15	14	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝑁, 𝑛} ∈ ran 𝐸) → 𝑁 ∈ 𝑉)
16	15	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝑉 USGrph 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 → 𝑁 ∈ 𝑉))
17	16	con3d 147	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → (¬ 𝑁 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
18	17	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑉 → (𝑉 USGrph 𝐸 → (¬ 𝑁 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
19	18	com13 86	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
20	19	a1d 25	. . . . . . . . . . 11 ⊢ (¬ 𝑁 ∈ 𝑉 → (¬ 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
21	13, 20	pm2.61i 175	. . . . . . . . . 10 ⊢ (¬ 𝑁 ∈ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
22	5, 21	sylbi 206	. . . . . . . . 9 ⊢ (𝑁 ∉ {𝑁, 𝑛} → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
23	4, 22	syl6 34	. . . . . . . 8 ⊢ ({𝑁, 𝑛} ∈ ran 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥 → (𝑉 USGrph 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
24	23	com13 86	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥 → ({𝑁, 𝑛} ∈ ran 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))))
25	24	imp 444	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → ({𝑁, 𝑛} ∈ ran 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸)))
26		ax-1 6	. . . . . 6 ⊢ (¬ {𝑁, 𝑛} ∈ ran 𝐸 → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
27	25, 26	pm2.61d1 170	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → (𝑛 ∈ 𝑉 → ¬ {𝑁, 𝑛} ∈ ran 𝐸))
28	27	ralrimiv 2948	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → ∀𝑛 ∈ 𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
29		rabeq0 3911	. . . 4 ⊢ ({𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅ ↔ ∀𝑛 ∈ 𝑉 ¬ {𝑁, 𝑛} ∈ ran 𝐸)
30	28, 29	sylibr 223	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = ∅)
31	2, 30	eqtrd 2644	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
32	31	ex 449	1 ⊢ (𝑉 USGrph 𝐸 → (∀𝑥 ∈ ran 𝐸 𝑁 ∉ 𝑥 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅))

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)