Theorem uvtx01vtx 27187

Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.)

Hypotheses

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)
isuvtx.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
uvtx01vtx	⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1))

Proof of Theorem uvtx01vtx

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxval 27177	. . . 4 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
3	2	a1i 11	. . 3 ⊢ (𝐸 = ∅ → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4	3	neeq1d 3046	. 2 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅))
5		rabn0 4293	. . 3 ⊢ ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))
6	5	a1i 11	. 2 ⊢ (𝐸 = ∅ → ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
7		falseral0 4417	. . . . . . . . . 10 ⊢ ((∀𝑛 ¬ 𝑛 ∈ ∅ ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → (𝑉 ∖ {𝑣}) = ∅)
8	7	ex 416	. . . . . . . . 9 ⊢ (∀𝑛 ¬ 𝑛 ∈ ∅ → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅))
9		noel 4247	. . . . . . . . 9 ⊢ ¬ 𝑛 ∈ ∅
10	8, 9	mpg 1799	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅)
11		ssdif0 4277	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 ∖ {𝑣}) = ∅)
12		sssn 4719	. . . . . . . . . 10 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑣}))
13		ne0i 4250	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → 𝑉 ≠ ∅)
14		eqneqall 2998	. . . . . . . . . . . 12 ⊢ (𝑉 = ∅ → (𝑉 ≠ ∅ → 𝑉 = {𝑣}))
15	13, 14	syl5 34	. . . . . . . . . . 11 ⊢ (𝑉 = ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
16		ax-1 6	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
17	15, 16	jaoi 854	. . . . . . . . . 10 ⊢ ((𝑉 = ∅ ∨ 𝑉 = {𝑣}) → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
18	12, 17	sylbi 220	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
19	11, 18	sylbir 238	. . . . . . . 8 ⊢ ((𝑉 ∖ {𝑣}) = ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
20	10, 19	syl 17	. . . . . . 7 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
21	20	impcom 411	. . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → 𝑉 = {𝑣})
22		vsnid 4562	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑣}
23		eleq2 2878	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝑣}))
24	22, 23	mpbiri 261	. . . . . . 7 ⊢ (𝑉 = {𝑣} → 𝑣 ∈ 𝑉)
25		ralel 3117	. . . . . . . 8 ⊢ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅
26		difeq1 4043	. . . . . . . . . 10 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ({𝑣} ∖ {𝑣}))
27		difid 4284	. . . . . . . . . 10 ⊢ ({𝑣} ∖ {𝑣}) = ∅
28	26, 27	eqtrdi 2849	. . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ∅)
29	28	raleqdv 3364	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅))
30	25, 29	mpbiri 261	. . . . . . 7 ⊢ (𝑉 = {𝑣} → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)
31	24, 30	jca 515	. . . . . 6 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
32	21, 31	impbii 212	. . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ 𝑉 = {𝑣})
33	32	a1i 11	. . . 4 ⊢ (𝐸 = ∅ → ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ 𝑉 = {𝑣}))
34	33	exbidv 1922	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ ∃𝑣 𝑉 = {𝑣}))
35		isuvtx.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
36	35	eqeq1i 2803	. . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
37		nbgr0edg 27147	. . . . . . 7 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
38	36, 37	sylbi 220	. . . . . 6 ⊢ (𝐸 = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
39	38	eleq2d 2875	. . . . 5 ⊢ (𝐸 = ∅ → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ ∅))
40	39	rexralbidv 3260	. . . 4 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
41		df-rex 3112	. . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
42	40, 41	syl6bb 290	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)))
43	1	fvexi 6659	. . . 4 ⊢ 𝑉 ∈ V
44		hash1snb 13776	. . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
45	43, 44	mp1i 13	. . 3 ⊢ (𝐸 = ∅ → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
46	34, 42, 45	3bitr4d 314	. 2 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (♯‘𝑉) = 1))
47	4, 6, 46	3bitrd 308	1 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525  ‘cfv 6324  (class class class)co 7135  1c1 10527  ♯chash 13686  Vtxcvtx 26789  Edgcedg 26840   NeighbVtx cnbgr 27122  UnivVtxcuvtx 27175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-nbgr 27123  df-uvtx 27176

This theorem is referenced by: (None)