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Theorem cusgredgex 35106
Description: Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
Hypotheses
Ref Expression
cusgredgex.1 𝑉 = (Vtx‘𝐺)
cusgredgex.2 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
cusgredgex (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))

Proof of Theorem cusgredgex
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 cusgrcplgr 29452 . . . . . . . 8 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2 cusgredgex.1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
3 cusgredgex.2 . . . . . . . . 9 𝐸 = (Edg‘𝐺)
42, 3cplgredgex 35105 . . . . . . . 8 (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
51, 4syl 17 . . . . . . 7 (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
65imp 406 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)
7 df-rex 3069 . . . . . 6 (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 ↔ ∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
86, 7sylib 218 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒))
9 eldifsni 4795 . . . . . . . . . . . . . . . 16 (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐵𝐴)
109necomd 2994 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐴𝐵)
1110adantl 481 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝐵)
12 hashprg 14431 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → (𝐴𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2))
1311, 12mpbid 232 . . . . . . . . . . . . 13 ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → (♯‘{𝐴, 𝐵}) = 2)
1413adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘{𝐴, 𝐵}) = 2)
15 cusgrusgr 29451 . . . . . . . . . . . . . 14 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
163usgredgppr 29228 . . . . . . . . . . . . . 14 ((𝐺 ∈ USGraph ∧ 𝑒𝐸) → (♯‘𝑒) = 2)
1715, 16sylan 580 . . . . . . . . . . . . 13 ((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) → (♯‘𝑒) = 2)
1817adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘𝑒) = 2)
1914, 18eqtr4d 2778 . . . . . . . . . . 11 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (♯‘{𝐴, 𝐵}) = (♯‘𝑒))
20 simpl 482 . . . . . . . . . . 11 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸))
21 vex 3482 . . . . . . . . . . . . . . . 16 𝑒 ∈ V
22 2nn0 12541 . . . . . . . . . . . . . . . 16 2 ∈ ℕ0
23 hashvnfin 14396 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ V ∧ 2 ∈ ℕ0) → ((♯‘𝑒) = 2 → 𝑒 ∈ Fin))
2421, 22, 23mp2an 692 . . . . . . . . . . . . . . 15 ((♯‘𝑒) = 2 → 𝑒 ∈ Fin)
2517, 24syl 17 . . . . . . . . . . . . . 14 ((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) → 𝑒 ∈ Fin)
26 fisshasheq 35099 . . . . . . . . . . . . . 14 ((𝑒 ∈ Fin ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
2725, 26syl3an1 1162 . . . . . . . . . . . . 13 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
28273comr 1124 . . . . . . . . . . . 12 (((♯‘{𝐴, 𝐵}) = (♯‘𝑒) ∧ (𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} = 𝑒)
29283exp 1118 . . . . . . . . . . 11 ((♯‘{𝐴, 𝐵}) = (♯‘𝑒) → ((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒)))
3019, 20, 29sylc 65 . . . . . . . . . 10 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
31303impa 1109 . . . . . . . . 9 ((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸 ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
32313com23 1125 . . . . . . . 8 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) ∧ 𝑒𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
33323expia 1120 . . . . . . 7 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (𝑒𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒)))
3433imdistand 570 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ((𝑒𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒) → (𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒)))
3534eximdv 1915 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → (∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒)))
368, 35mpd 15 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒))
37 pm3.22 459 . . . . . 6 ((𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒) → ({𝐴, 𝐵} = 𝑒𝑒𝐸))
38 eqcom 2742 . . . . . . 7 ({𝐴, 𝐵} = 𝑒𝑒 = {𝐴, 𝐵})
3938anbi1i 624 . . . . . 6 (({𝐴, 𝐵} = 𝑒𝑒𝐸) ↔ (𝑒 = {𝐴, 𝐵} ∧ 𝑒𝐸))
4037, 39sylib 218 . . . . 5 ((𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒) → (𝑒 = {𝐴, 𝐵} ∧ 𝑒𝐸))
4140eximi 1832 . . . 4 (∃𝑒(𝑒𝐸 ∧ {𝐴, 𝐵} = 𝑒) → ∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒𝐸))
4236, 41syl 17 . . 3 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒𝐸))
43 prex 5443 . . . 4 {𝐴, 𝐵} ∈ V
44 eleq1 2827 . . . 4 (𝑒 = {𝐴, 𝐵} → (𝑒𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
4543, 44ceqsexv 3530 . . 3 (∃𝑒(𝑒 = {𝐴, 𝐵} ∧ 𝑒𝐸) ↔ {𝐴, 𝐵} ∈ 𝐸)
4642, 45sylib 218 . 2 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → {𝐴, 𝐵} ∈ 𝐸)
4746ex 412 1 (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  {csn 4631  {cpr 4633  cfv 6563  Fincfn 8984  2c2 12319  0cn0 12524  chash 14366  Vtxcvtx 29028  Edgcedg 29079  USGraphcusgr 29181  ComplGraphccplgr 29441  ComplUSGraphccusgr 29442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-usgr 29183  df-nbgr 29365  df-uvtx 29418  df-cplgr 29443  df-cusgr 29444
This theorem is referenced by:  cusgredgex2  35107
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