Theorem nbusgrvtxm1 29325

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 770	. . . . . . 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 3036	. . . . . . . . . 10 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
9	8	biimpri 228	. . . . . . . . 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 29324	. . . . . . 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 1373	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15		breq1 5126	. . . . . . . . 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 29265	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18		hashcl 14378	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
19		nn0re 12518	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
20		1red 11244	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21		2re 12322	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23		id 22	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24		1lt2 12419	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → 1 < 2)
26	20, 22, 23, 25	ltsub2dd 11858	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
27	23, 22	resubcld 11673	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28		peano2rem 11558	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
29	27, 28	ltnled 11390	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℝ → (((♯‘𝑉) − 2) < ((♯‘𝑉) − 1) ↔ ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
30	26, 29	mpbid 232	. . . . . . . . . . . 12 ⊢ ((♯‘𝑉) ∈ ℝ → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
31	17, 18, 19, 30	4syl 19	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FinUSGraph → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
32	31	pm2.21d 121	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
34	33	ad3antlr 731	. . . . . . . 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)))
35	16, 34	sylbid 240	. . . . . . 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)))
36	35	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))))
37	14, 36	mpid 44	. . . . 5 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
38	37	ex 412	. . . 4 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
39	38	com23 86	. . 3 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
40	39	ex 412	. 2 ⊢ (¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
41	2, 40	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 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   ∉ wnel 3035   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  Fincfn 8967  ℝcr 11136  1c1 11138   < clt 11277   ≤ cle 11278   − cmin 11474  2c2 12303  ℕ₀cn0 12509  ♯chash 14352  Vtxcvtx 28942  FinUSGraphcfusgr 29262   NeighbVtx cnbgr 29278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-oadd 8492  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-dju 9923  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-n0 12510  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13530  df-hash 14353  df-fusgr 29263  df-nbgr 29279

This theorem is referenced by:  nbusgrvtxm1uvtx  29351