Theorem nbusgrvtxm1 40607

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 476	. . . . . . . 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 790	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∈ 𝑉)
6		simpr 476	. . . . . . . 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 2783	. . . . . . . . . 10 ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
9	8	biimpri 217	. . . . . . . . 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 40606	. . . . . . 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 1320	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
15		breq1 4586	. . . . . . . . 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 40538	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18		hashcl 13009	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
19		nn0re 11178	. . . . . . . . . . . . 13 ⊢ ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℝ)
20		1red 9934	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21		2re 10967	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23		id 22	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → (#‘𝑉) ∈ ℝ)
24		1lt2 11071	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → 1 < 2)
26	20, 22, 23, 25	ltsub2dd 10519	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) < ((#‘𝑉) − 1))
27	23, 22	resubcld 10337	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) ∈ ℝ)
28		peano2rem 10227	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 1) ∈ ℝ)
29	27, 28	ltnled 10063	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑉) ∈ ℝ → (((#‘𝑉) − 2) < ((#‘𝑉) − 1) ↔ ¬ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
30	26, 29	mpbid 221	. . . . . . . . . . . . 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 117	. . . . . . . . . 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 763	. . . . . . . 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 229	. . . . . . 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 449	. . . . . 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 43	. . . . 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 449	. . . 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 84	. . 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 449	. 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 175	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝcr 9814  1c1 9816   < clt 9953   ≤ cle 9954   − cmin 10145  2c2 10947  ℕ₀cn0 11169  #chash 12979  Vtxcvtx 25673   FinUSGraph cfusgr 40535   NeighbVtx cnbgr 40550

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-rep 4699  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-fal 1481  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-rmo 2904  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-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  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-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-fusgr 40536  df-nbgr 40554

This theorem is referenced by:  nbusgrvtxm1uvtx  40632