Theorem uvtx01vtx 27331

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 27321	. . . 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 2993	. 2 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅))
5		rabn0 4271	. . 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 4403	. . . . . . . . . 10 ⊢ ((∀𝑛 ¬ 𝑛 ∈ ∅ ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → (𝑉 ∖ {𝑣}) = ∅)
8	7	ex 416	. . . . . . . . 9 ⊢ (∀𝑛 ¬ 𝑛 ∈ ∅ → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅))
9		noel 4217	. . . . . . . . 9 ⊢ ¬ 𝑛 ∈ ∅
10	8, 9	mpg 1804	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅)
11		ssdif0 4250	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 ∖ {𝑣}) = ∅)
12		sssn 4711	. . . . . . . . . 10 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑣}))
13		ne0i 4221	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → 𝑉 ≠ ∅)
14		eqneqall 2945	. . . . . . . . . . . 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 856	. . . . . . . . . 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 4550	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑣}
23		eleq2 2821	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝑣}))
24	22, 23	mpbiri 261	. . . . . . 7 ⊢ (𝑉 = {𝑣} → 𝑣 ∈ 𝑉)
25		ralel 3064	. . . . . . . 8 ⊢ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅
26		difeq1 4004	. . . . . . . . . 10 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ({𝑣} ∖ {𝑣}))
27		difid 4257	. . . . . . . . . 10 ⊢ ({𝑣} ∖ {𝑣}) = ∅
28	26, 27	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ∅)
29	28	raleqdv 3315	. . . . . . . 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 1927	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ ∃𝑣 𝑉 = {𝑣}))
35		isuvtx.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
36	35	eqeq1i 2743	. . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
37		nbgr0edg 27291	. . . . . . 7 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
38	36, 37	sylbi 220	. . . . . 6 ⊢ (𝐸 = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
39	38	eleq2d 2818	. . . . 5 ⊢ (𝐸 = ∅ → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ ∅))
40	39	rexralbidv 3210	. . . 4 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
41		df-rex 3059	. . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
42	40, 41	bitrdi 290	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)))
43	1	fvexi 6682	. . . 4 ⊢ 𝑉 ∈ V
44		hash1snb 13865	. . . 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 846  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2113   ≠ wne 2934  ∀wral 3053  ∃wrex 3054  {crab 3057  Vcvv 3397   ∖ cdif 3838   ⊆ wss 3841  ∅c0 4209  {csn 4513  ‘cfv 6333  (class class class)co 7164  1c1 10609  ♯chash 13775  Vtxcvtx 26933  Edgcedg 26984   NeighbVtx cnbgr 27266  UnivVtxcuvtx 27319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  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 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-oadd 8128  df-er 8313  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-dju 9396  df-card 9434  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-hash 13776  df-nbgr 27267  df-uvtx 27320

This theorem is referenced by: (None)