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Theorem nbusgrvtxm1 26864
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 473 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉))
5 simprl 758 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑉)
6 simpr 477 . . . . . . . 8 ((𝑀𝑉𝑀𝑈) → 𝑀𝑈)
76adantl 474 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀𝑈)
8 df-nel 3074 . . . . . . . . . 10 (𝑀 ∉ (𝐺 NeighbVtx 𝑈) ↔ ¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈))
98biimpri 220 . . . . . . . . 9 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
109adantr 473 . . . . . . . 8 ((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
1110adantr 473 . . . . . . 7 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → 𝑀 ∉ (𝐺 NeighbVtx 𝑈))
12 hashnbusgrnn0.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
1312nbfusgrlevtxm2 26863 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
144, 5, 7, 11, 13syl13anc 1352 . . . . . 6 (((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) → (♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2))
15 breq1 4932 . . . . . . . . 9 ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
1615adantl 474 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) ↔ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
1712fusgrvtxfi 26804 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
18 hashcl 13532 . . . . . . . . . . . 12 (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
19 nn0re 11717 . . . . . . . . . . . . 13 ((♯‘𝑉) ∈ ℕ0 → (♯‘𝑉) ∈ ℝ)
20 1red 10440 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → 1 ∈ ℝ)
21 2re 11514 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
2221a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → 2 ∈ ℝ)
23 id 22 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → (♯‘𝑉) ∈ ℝ)
24 1lt2 11618 . . . . . . . . . . . . . . . 16 1 < 2
2524a1i 11 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → 1 < 2)
2620, 22, 23, 25ltsub2dd 11054 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) < ((♯‘𝑉) − 1))
2723, 22resubcld 10869 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 2) ∈ ℝ)
28 peano2rem 10754 . . . . . . . . . . . . . . 15 ((♯‘𝑉) ∈ ℝ → ((♯‘𝑉) − 1) ∈ ℝ)
2927, 28ltnled 10587 . . . . . . . . . . . . . 14 ((♯‘𝑉) ∈ ℝ → (((♯‘𝑉) − 2) < ((♯‘𝑉) − 1) ↔ ¬ ((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2)))
3026, 29mpbid 224 . . . . . . . . . . . . 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 119 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3433adantr 473 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3534ad3antlr 718 . . . . . . . 8 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → (((♯‘𝑉) − 1) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3616, 35sylbid 232 . . . . . . 7 ((((¬ 𝑀 ∈ (𝐺 NeighbVtx 𝑈) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑈𝑉)) ∧ (𝑀𝑉𝑀𝑈)) ∧ (♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1)) → ((♯‘(𝐺 NeighbVtx 𝑈)) ≤ ((♯‘𝑉) − 2) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))
3736ex 405 . . . . . 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 405 . . . 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 405 . 2 𝑀 ∈ (𝐺 NeighbVtx 𝑈) → ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((♯‘(𝐺 NeighbVtx 𝑈)) = ((♯‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈)))))
422, 41pm2.61i 177 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 198  wa 387   = wceq 1507  wcel 2050  wne 2967  wnel 3073   class class class wbr 4929  cfv 6188  (class class class)co 6976  Fincfn 8306  cr 10334  1c1 10336   < clt 10474  cle 10475  cmin 10670  2c2 11495  0cn0 11707  chash 13505  Vtxcvtx 26484  FinUSGraphcfusgr 26801   NeighbVtx cnbgr 26817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-dju 9124  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-n0 11708  df-xnn0 11780  df-z 11794  df-uz 12059  df-fz 12709  df-hash 13506  df-fusgr 26802  df-nbgr 26818
This theorem is referenced by:  nbusgrvtxm1uvtx  26890
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