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Theorem fusgrmaxsize 27261
 Description: The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
Hypotheses
Ref Expression
fusgrmaxsize.v 𝑉 = (Vtx‘𝐺)
fusgrmaxsize.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
fusgrmaxsize (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))

Proof of Theorem fusgrmaxsize
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 fusgrmaxsize.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 27115 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 cusgrexg 27241 . . . 4 (𝑉 ∈ Fin → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
43adantl 485 . . 3 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
5 fusgrmaxsize.e . . . . . 6 𝐸 = (Edg‘𝐺)
61fvexi 6659 . . . . . . . 8 𝑉 ∈ V
7 vex 3444 . . . . . . . 8 𝑒 ∈ V
86, 7opvtxfvi 26809 . . . . . . 7 (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉
98eqcomi 2807 . . . . . 6 𝑉 = (Vtx‘⟨𝑉, 𝑒⟩)
10 eqid 2798 . . . . . 6 (Edg‘⟨𝑉, 𝑒⟩) = (Edg‘⟨𝑉, 𝑒⟩)
111, 5, 9, 10sizusglecusg 27260 . . . . 5 ((𝐺 ∈ USGraph ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
1211adantlr 714 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
139, 10cusgrsize 27251 . . . . . . . 8 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2))
14 breq2 5034 . . . . . . . . 9 ((♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) ↔ (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1514biimpd 232 . . . . . . . 8 ((♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1613, 15syl 17 . . . . . . 7 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1716expcom 417 . . . . . 6 (𝑉 ∈ Fin → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))))
1817adantl 485 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))))
1918imp 410 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
2012, 19mpd 15 . . 3 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
214, 20exlimddv 1936 . 2 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
222, 21sylbi 220 1 (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Fincfn 8494   ≤ cle 10667  2c2 11682  Ccbc 13660  ♯chash 13688  Vtxcvtx 26796  Edgcedg 26847  USGraphcusgr 26949  FinUSGraphcfusgr 27113  ComplUSGraphccusgr 27207 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-dju 9316  df-card 9354  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-n0 11888  df-xnn0 11958  df-z 11972  df-uz 12234  df-rp 12380  df-fz 12888  df-seq 13367  df-fac 13632  df-bc 13661  df-hash 13689  df-vtx 26798  df-iedg 26799  df-edg 26848  df-uhgr 26858  df-upgr 26882  df-umgr 26883  df-uspgr 26950  df-usgr 26951  df-fusgr 27114  df-nbgr 27130  df-uvtx 27183  df-cplgr 27208  df-cusgr 27209 This theorem is referenced by: (None)
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