Theorem nbusgrvtxm1 28327

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 485	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
4	3	adantr 481	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉))
5		simprl 769	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ∈ 𝑉)
6		simpr 485	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈) → 𝑀 ≠ 𝑈)
7	6	adantl 482	. . . . . . 7 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → 𝑀 ≠ 𝑈)
8		df-nel 3050	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
11	10	adantr 481	. . . . . . 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 28326	. . . . . . 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 1372	. . . . . 6 ⊢ (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉)) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15		breq1 5108	. . . . . . . . 9 ⊢ ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
16	15	adantl 482	. . . . . . . 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 28267	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18		hashcl 14256	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
19		nn0re 12422	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
20		1red 11156	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21		2re 12227	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23		id 22	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24		1lt2 12324	. . . . . . . . . . . . . . . 16 ⊢ 1 < 2
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → 1 < 2)
26	20, 22, 23, 25	ltsub2dd 11768	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
27	23, 22	resubcld 11583	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28		peano2rem 11468	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
29	27, 28	ltnled 11302	. . . . . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑈 ∈ 𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
35	34	ad3antlr 729	. . . . . . . 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 413	. . . . . 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 413	. . . 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 413	. 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 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  ℝcr 11050  1c1 11052   < clt 11189   ≤ cle 11190   − cmin 11385  2c2 12208  ℕ₀cn0 12413  ♯chash 14230  Vtxcvtx 27947  FinUSGraphcfusgr 28264   NeighbVtx cnbgr 28280

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-fusgr 28265  df-nbgr 28281

This theorem is referenced by:  nbusgrvtxm1uvtx  28353