Theorem nbusgrvtxm1 29060

Description: If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
hashnbusgrnn0.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
nbusgrvtxm1	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))

Proof of Theorem nbusgrvtxm1

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
2	1	2a1d 26	. 2 ⊢ (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
3		simpr 484	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
4	3	adantr 480	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
5		simprl 768	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∈ 𝑉)
6		simpr 484	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ≠ 𝑈)
7	6	adantl 481	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ≠ 𝑈)
8		df-nel 3039	. . . . . . . . . 10 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
9	8	biimpri 227	. . . . . . . . 9 ⊢ (¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
10	9	adantr 480	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
11	10	adantr 480	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
12		hashnbusgrnn0.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	nbfusgrlevtxm2 29059	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈 ∧ 𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
14	4, 5, 7, 11, 13	syl13anc 1369	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15		breq1 5141	. . . . . . . . 9 ⊢ ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
17	12	fusgrvtxfi 29000	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18		hashcl 14312	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
19		nn0re 12477	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
20		1red 11211	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21		2re 12282	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23		id 22	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24		1lt2 12379	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 < 2)
26	20, 22, 23, 25	ltsub2dd 11823	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
27	23, 22	resubcld 11638	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28		peano2rem 11523	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
29	27, 28	ltnled 11357	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → (((♯‘𝑉) − 2) < ((♯‘𝑉) − 1) ↔ ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
30	26, 29	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℝ → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
31	19, 30	syl 17	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) ∈ ℕ0 → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
32	17, 18, 31	3syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FinUSGraph → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
33	32	pm2.21d 121	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
35	34	ad3antlr 728	. . . . . . . 8 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
36	16, 35	sylbid 239	. . . . . . 7 ⊢ ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
37	36	ex 412	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
38	14, 37	mpid 44	. . . . 5 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
39	38	ex 412	. . . 4 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
40	39	com23 86	. . 3 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
41	40	ex 412	. 2 ⊢ (¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
42	2, 41	pm2.61i 182	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2932   ∉ wnel 3038   class class class wbr 5138  ‘cfv 6533  (class class class)co 7401  Fincfn 8934  ℝcr 11104  1c1 11106   < clt 11244   ≤ cle 11245   − cmin 11440  2c2 12263  ℕ₀cn0 12468  ♯chash 14286  Vtxcvtx 28680  FinUSGraphcfusgr 28997   NeighbVtx cnbgr 29013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-fusgr 28998  df-nbgr 29014

This theorem is referenced by:  nbusgrvtxm1uvtx  29086