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Theorem fusgrmaxsize 26591
 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 26430 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 cusgrexg 26571 . . . 4 (𝑉 ∈ Fin → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
43adantl 473 . . 3 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
5 fusgrmaxsize.e . . . . . 6 𝐸 = (Edg‘𝐺)
6 fvex 6363 . . . . . . . . 9 (Vtx‘𝐺) ∈ V
71, 6eqeltri 2835 . . . . . . . 8 𝑉 ∈ V
8 vex 3343 . . . . . . . 8 𝑒 ∈ V
9 opvtxfv 26104 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉)
107, 8, 9mp2an 710 . . . . . . 7 (Vtx‘⟨𝑉, 𝑒⟩) = 𝑉
1110eqcomi 2769 . . . . . 6 𝑉 = (Vtx‘⟨𝑉, 𝑒⟩)
12 eqid 2760 . . . . . 6 (Edg‘⟨𝑉, 𝑒⟩) = (Edg‘⟨𝑉, 𝑒⟩)
131, 5, 11, 12sizusglecusg 26590 . . . . 5 ((𝐺 ∈ USGraph ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
1413adantlr 753 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)))
1511, 12cusgrsize 26581 . . . . . . . 8 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2))
16 breq2 4808 . . . . . . . . 9 ((♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) ↔ (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1716biimpd 219 . . . . . . . 8 ((♯‘(Edg‘⟨𝑉, 𝑒⟩)) = ((♯‘𝑉)C2) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1815, 17syl 17 . . . . . . 7 ((⟨𝑉, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
1918expcom 450 . . . . . 6 (𝑉 ∈ Fin → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))))
2019adantl 473 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (⟨𝑉, 𝑒⟩ ∈ ComplUSGraph → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))))
2120imp 444 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → ((♯‘𝐸) ≤ (♯‘(Edg‘⟨𝑉, 𝑒⟩)) → (♯‘𝐸) ≤ ((♯‘𝑉)C2)))
2214, 21mpd 15 . . 3 (((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) ∧ ⟨𝑉, 𝑒⟩ ∈ ComplUSGraph) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
234, 22exlimddv 2012 . 2 ((𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
242, 23sylbi 207 1 (𝐺 ∈ FinUSGraph → (♯‘𝐸) ≤ ((♯‘𝑉)C2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   class class class wbr 4804  ‘cfv 6049  (class class class)co 6814  Fincfn 8123   ≤ cle 10287  2c2 11282  Ccbc 13303  ♯chash 13331  Vtxcvtx 26094  Edgcedg 26159  USGraphcusgr 26264  FinUSGraphcfusgr 26428  ComplUSGraphccusgr 26536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-seq 13016  df-fac 13275  df-bc 13304  df-hash 13332  df-vtx 26096  df-iedg 26097  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-umgr 26198  df-uspgr 26265  df-usgr 26266  df-fusgr 26429  df-nbgr 26445  df-uvtx 26507  df-cplgr 26537  df-cusgr 26538 This theorem is referenced by: (None)
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