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Theorem cusgredgex 35127
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 29437 . . . . . . . 8 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2 cusgredgex.1 . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
3 cusgredgex.2 . . . . . . . . 9 𝐸 = (Edg‘𝐺)
42, 3cplgredgex 35126 . . . . . . . 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 3071 . . . . . 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 4790 . . . . . . . . . . . . . . . 16 (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐵𝐴)
109necomd 2996 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝑉 ∖ {𝐴}) → 𝐴𝐵)
1110adantl 481 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → 𝐴𝐵)
12 hashprg 14434 . . . . . . . . . . . . . 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 29436 . . . . . . . . . . . . . 14 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
163usgredgppr 29213 . . . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . 16 𝑒 ∈ V
22 2nn0 12543 . . . . . . . . . . . . . . . 16 2 ∈ ℕ0
23 hashvnfin 14399 . . . . . . . . . . . . . . . 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 35120 . . . . . . . . . . . . . 14 ((𝑒 ∈ Fin ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
2725, 26syl3an1 1164 . . . . . . . . . . . . 13 (((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒 ∧ (♯‘{𝐴, 𝐵}) = (♯‘𝑒)) → {𝐴, 𝐵} = 𝑒)
28273comr 1126 . . . . . . . . . . . 12 (((♯‘{𝐴, 𝐵}) = (♯‘𝑒) ∧ (𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} = 𝑒)
29283exp 1120 . . . . . . . . . . 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 1110 . . . . . . . . 9 ((𝐺 ∈ ComplUSGraph ∧ 𝑒𝐸 ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴}))) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
32313com23 1127 . . . . . . . 8 ((𝐺 ∈ ComplUSGraph ∧ (𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) ∧ 𝑒𝐸) → ({𝐴, 𝐵} ⊆ 𝑒 → {𝐴, 𝐵} = 𝑒))
33323expia 1122 . . . . . . 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 1917 . . . . 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 2744 . . . . . . 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 1835 . . . 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 5437 . . . 4 {𝐴, 𝐵} ∈ V
44 eleq1 2829 . . . 4 (𝑒 = {𝐴, 𝐵} → (𝑒𝐸 ↔ {𝐴, 𝐵} ∈ 𝐸))
4543, 44ceqsexv 3532 . . 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 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  {csn 4626  {cpr 4628  cfv 6561  Fincfn 8985  2c2 12321  0cn0 12526  chash 14369  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166  ComplGraphccplgr 29426  ComplUSGraphccusgr 29427
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-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-edg 29065  df-usgr 29168  df-nbgr 29350  df-uvtx 29403  df-cplgr 29428  df-cusgr 29429
This theorem is referenced by:  cusgredgex2  35128
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