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Theorem nbusgrvtxm1 26262
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 477 . . . . . . . 8 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉))
43adantr 481 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉))
5 simprl 793 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑉)
6 simpr 477 . . . . . . . 8 ((𝑀𝑉𝑀𝑈) → 𝑀𝑈)
76adantl 482 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑈)
8 df-nel 2895 . . . . . . . . . 10 (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
98biimpri 218 . . . . . . . . 9 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
109adantr 481 . . . . . . . 8 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
1110adantr 481 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
12 hashnbusgrnn0.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1312nbfusgrlevtxm2 26261 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
144, 5, 7, 11, 13syl13anc 1326 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
15 breq1 4647 . . . . . . . . 9 ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) ↔ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
1615adantl 482 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1)) → ((#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2) ↔ ((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2)))
1712fusgrvtxfi 26192 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18 hashcl 13130 . . . . . . . . . . . 12 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
19 nn0re 11286 . . . . . . . . . . . . 13 ((#‘𝑉) ∈ ℕ0 → (#‘𝑉) ∈ ℝ)
20 1red 10040 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21 2re 11075 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23 id 22 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → (#‘𝑉) ∈ ℝ)
24 1lt2 11179 . . . . . . . . . . . . . . . 16 1 < 2
2524a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → 1 < 2)
2620, 22, 23, 25ltsub2dd 10625 . . . . . . . . . . . . . 14 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) < ((#‘𝑉) − 1))
2723, 22resubcld 10443 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 2) ∈ ℝ)
28 peano2rem 10333 . . . . . . . . . . . . . . 15 ((#‘𝑉) ∈ ℝ → ((#‘𝑉) − 1) ∈ ℝ)
2927, 28ltnled 10169 . . . . . . . . . . . . . 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 481 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (((#‘𝑉) − 1) ≤ ((#‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3534ad3antlr 766 . . . . . . . 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 450 . . . . . 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 450 . . . 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 450 . 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 384   = wceq 1481  wcel 1988  wne 2791  wnel 2894   class class class wbr 4644  cfv 5876  (class class class)co 6635  Fincfn 7940  cr 9920  1c1 9922   < clt 10059  cle 10060  cmin 10251  2c2 11055  0cn0 11277  #chash 13100  Vtxcvtx 25855   FinUSGraph cfusgr 26189   NeighbVtx cnbgr 26205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-fusgr 26190  df-nbgr 26209
This theorem is referenced by:  nbusgrvtxm1uvtx  26287
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