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Theorem cusgredg 29354
Description: In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.)
Hypotheses
Ref Expression
iscusgrvtx.v 𝑉 = (Vtx‘𝐺)
iscusgredg.v 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
cusgredg (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem cusgredg
Dummy variables 𝑣 𝑛 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscusgrvtx.v . . 3 𝑉 = (Vtx‘𝐺)
2 iscusgredg.v . . 3 𝐸 = (Edg‘𝐺)
31, 2iscusgredg 29353 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸))
4 usgredgss 29089 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
51pweqi 4613 . . . . . 6 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
65rabeqi 3433 . . . . 5 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}
74, 2, 63sstr4g 4024 . . . 4 (𝐺 ∈ USGraph → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
87adantr 479 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
9 elss2prb 14498 . . . . 5 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}))
10 sneq 4633 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑣} = {𝑦})
1110difeq2d 4118 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑦}))
12 preq2 4733 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑛, 𝑣} = {𝑛, 𝑦})
1312eleq1d 2811 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ({𝑛, 𝑣} ∈ 𝐸 ↔ {𝑛, 𝑦} ∈ 𝐸))
1411, 13raleqbidv 3330 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (∀𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1514rspcv 3603 . . . . . . . . . . . 12 (𝑦𝑉 → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1615adantr 479 . . . . . . . . . . 11 ((𝑦𝑉𝑧𝑉) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1716adantr 479 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
18 simpr 483 . . . . . . . . . . . . 13 ((𝑦𝑉𝑧𝑉) → 𝑧𝑉)
19 necom 2984 . . . . . . . . . . . . . . 15 (𝑦𝑧𝑧𝑦)
2019biimpi 215 . . . . . . . . . . . . . 14 (𝑦𝑧𝑧𝑦)
2120adantr 479 . . . . . . . . . . . . 13 ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑧𝑦)
2218, 21anim12i 611 . . . . . . . . . . . 12 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝑧𝑉𝑧𝑦))
23 eldifsn 4785 . . . . . . . . . . . 12 (𝑧 ∈ (𝑉 ∖ {𝑦}) ↔ (𝑧𝑉𝑧𝑦))
2422, 23sylibr 233 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → 𝑧 ∈ (𝑉 ∖ {𝑦}))
25 preq1 4732 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛, 𝑦} = {𝑧, 𝑦})
2625eleq1d 2811 . . . . . . . . . . . 12 (𝑛 = 𝑧 → ({𝑛, 𝑦} ∈ 𝐸 ↔ {𝑧, 𝑦} ∈ 𝐸))
2726rspcv 3603 . . . . . . . . . . 11 (𝑧 ∈ (𝑉 ∖ {𝑦}) → (∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸 → {𝑧, 𝑦} ∈ 𝐸))
2824, 27syl 17 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸 → {𝑧, 𝑦} ∈ 𝐸))
29 id 22 . . . . . . . . . . . . . . . 16 (𝑝 = {𝑦, 𝑧} → 𝑝 = {𝑦, 𝑧})
30 prcom 4731 . . . . . . . . . . . . . . . 16 {𝑦, 𝑧} = {𝑧, 𝑦}
3129, 30eqtr2di 2783 . . . . . . . . . . . . . . 15 (𝑝 = {𝑦, 𝑧} → {𝑧, 𝑦} = 𝑝)
3231eleq1d 2811 . . . . . . . . . . . . . 14 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3332biimpd 228 . . . . . . . . . . . . 13 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3433a1d 25 . . . . . . . . . . . 12 (𝑝 = {𝑦, 𝑧} → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3534ad2antll 727 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3635com23 86 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → ({𝑧, 𝑦} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3717, 28, 363syld 60 . . . . . . . . 9 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3837ex 411 . . . . . . . 8 ((𝑦𝑉𝑧𝑉) → ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸))))
3938rexlimivv 3190 . . . . . . 7 (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
4039com13 88 . . . . . 6 (𝐺 ∈ USGraph → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸)))
4140imp 405 . . . . 5 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸))
429, 41biimtrid 241 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝑝𝐸))
4342ssrdv 3984 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ 𝐸)
448, 43eqssd 3996 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
453, 44sylbi 216 1 (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  cdif 3943  wss 3946  𝒫 cpw 4597  {csn 4623  {cpr 4625  cfv 6543  2c2 12310  chash 14339  Vtxcvtx 28926  Edgcedg 28977  USGraphcusgr 29079  ComplUSGraphccusgr 29340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-dju 9934  df-card 9972  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12256  df-2 12318  df-n0 12516  df-xnn0 12588  df-z 12602  df-uz 12866  df-fz 13530  df-hash 14340  df-edg 28978  df-upgr 29012  df-umgr 29013  df-usgr 29081  df-nbgr 29263  df-uvtx 29316  df-cplgr 29341  df-cusgr 29342
This theorem is referenced by:  cusgrfilem1  29386
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