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Theorem nbusgrvtxm1 26325
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 ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))

Proof of Theorem nbusgrvtxm1
StepHypRef Expression
1 ax-1 6 . . 3 (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
212a1d 26 . 2 (𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
3 simpr 476 . . . . . . . 8 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉))
43adantr 480 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉))
5 simprl 809 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑉)
6 simpr 476 . . . . . . . 8 ((𝑀𝑉𝑀𝑈) → 𝑀𝑈)
76adantl 481 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑈)
8 df-nel 2927 . . . . . . . . . 10 (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
98biimpri 218 . . . . . . . . 9 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
109adantr 480 . . . . . . . 8 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
1110adantr 480 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
12 hashnbusgrnn0.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1312nbfusgrlevtxm2 26324 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
144, 5, 7, 11, 13syl13anc 1368 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
15 breq1 4688 . . . . . . . . 9 ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) ↔ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
1615adantl 481 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1)) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) ↔ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
1712fusgrvtxfi 26256 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18 hashcl 13185 . . . . . . . . . . . 12 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
19 nn0re 11339 . . . . . . . . . . . . 13 ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℝ)
20 1red 10093 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21 2re 11128 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23 id 22 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → (#‘𝑉) ∈ ℝ)
24 1lt2 11232 . . . . . . . . . . . . . . . 16 1 < 2
2524a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 1 < 2)
2620, 22, 23, 25ltsub2dd 10678 . . . . . . . . . . . . . 14 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) < ((#‘𝑉) − 1))
2723, 22resubcld 10496 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) ∈ ℝ)
28 peano2rem 10386 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 1) ∈ ℝ)
2927, 28ltnled 10222 . . . . . . . . . . . . . 14 ((#‘𝑉) ∈ ℝ → (((#‘𝑉) − 2) < ((#‘𝑉) − 1) ↔ ¬ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
3026, 29mpbid 222 . . . . . . . . . . . . 13 ((#‘𝑉) ∈ ℝ → ¬ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2))
3119, 30syl 17 . . . . . . . . . . . 12 ((#‘𝑉) ∈ ℕ0 → ¬ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2))
3217, 18, 313syl 18 . . . . . . . . . . 11 (𝐺 ∈ FinUSGraph → ¬ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2))
3332pm2.21d 118 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → (((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3433adantr 480 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3534ad3antlr 767 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1)) → (((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3616, 35sylbid 230 . . . . . . 7 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1)) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3736ex 449 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
3814, 37mpid 44 . . . . 5 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3938ex 449 . . . 4 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → ((𝑀𝑉𝑀𝑈) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
4039com23 86 . . 3 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
4140ex 449 . 2 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
422, 41pm2.61i 176 1 ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wnel 2926   class class class wbr 4685  cfv 5926  (class class class)co 6690  Fincfn 7997  cr 9973  1c1 9975   < clt 10112  cle 10113  cmin 10304  2c2 11108  0cn0 11330  #chash 13157  Vtxcvtx 25919  FinUSGraphcfusgr 26253   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-fusgr 26254  df-nbgr 26270
This theorem is referenced by:  nbusgrvtxm1uvtx  26356
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