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Theorem nbusgrvtxm1 29396
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 484 . . . . . . . 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 771 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑉)
6 simpr 484 . . . . . . . 8 ((𝑀𝑉𝑀𝑈) → 𝑀𝑈)
76adantl 481 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑈)
8 df-nel 3047 . . . . . . . . . 10 (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
98biimpri 228 . . . . . . . . 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 29395 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
144, 5, 7, 11, 13syl13anc 1374 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15 breq1 5146 . . . . . . . . 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 29336 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18 hashcl 14395 . . . . . . . . . . . 12 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
19 nn0re 12535 . . . . . . . . . . . 12 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
20 1red 11262 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21 2re 12340 . . . . . . . . . . . . . . 15 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23 id 22 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24 1lt2 12437 . . . . . . . . . . . . . . 15 1 < 2
2524a1i 11 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → 1 < 2)
2620, 22, 23, 25ltsub2dd 11876 . . . . . . . . . . . . 13 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
2723, 22resubcld 11691 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28 peano2rem 11576 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
2927, 28ltnled 11408 . . . . . . . . . . . . 13 ((♯‘𝑉) ∈ ℝ → (((♯‘𝑉) − 2) < ((♯‘𝑉) − 1) ↔ ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
3026, 29mpbid 232 . . . . . . . . . . . 12 ((♯‘𝑉) ∈ ℝ → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
3117, 18, 19, 304syl 19 . . . . . . . . . . 11 (𝐺 ∈ FinUSGraph → ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2))
3231pm2.21d 121 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3332adantr 480 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3433ad3antlr 731 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3516, 34sylbid 240 . . . . . . 7 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3635ex 412 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
3714, 36mpid 44 . . . . 5 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3837ex 412 . . . 4 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → ((𝑀𝑉𝑀𝑈) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
3938com23 86 . . 3 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
4039ex 412 . 2 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
412, 40pm2.61i 182 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 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wnel 3046   class class class wbr 5143  cfv 6561  (class class class)co 7431  Fincfn 8985  cr 11154  1c1 11156   < clt 11295  cle 11296  cmin 11492  2c2 12321  0cn0 12526  chash 14369  Vtxcvtx 29013  FinUSGraphcfusgr 29333   NeighbVtx cnbgr 29349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-fusgr 29334  df-nbgr 29350
This theorem is referenced by:  nbusgrvtxm1uvtx  29422
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