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Theorem cusgredg 28414
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 28413 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸))
4 usgredgss 28152 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
51pweqi 4581 . . . . . 6 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
65rabeqi 3423 . . . . 5 {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}
74, 2, 63sstr4g 3994 . . . 4 (𝐺 ∈ USGraph → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
87adantr 482 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
9 elss2prb 14393 . . . . 5 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}))
10 sneq 4601 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑣} = {𝑦})
1110difeq2d 4087 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑦}))
12 preq2 4700 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑛, 𝑣} = {𝑛, 𝑦})
1312eleq1d 2823 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ({𝑛, 𝑣} ∈ 𝐸 ↔ {𝑛, 𝑦} ∈ 𝐸))
1411, 13raleqbidv 3322 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (∀𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1514rspcv 3580 . . . . . . . . . . . 12 (𝑦𝑉 → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1615adantr 482 . . . . . . . . . . 11 ((𝑦𝑉𝑧𝑉) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1716adantr 482 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
18 simpr 486 . . . . . . . . . . . . 13 ((𝑦𝑉𝑧𝑉) → 𝑧𝑉)
19 necom 2998 . . . . . . . . . . . . . . 15 (𝑦𝑧𝑧𝑦)
2019biimpi 215 . . . . . . . . . . . . . 14 (𝑦𝑧𝑧𝑦)
2120adantr 482 . . . . . . . . . . . . 13 ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑧𝑦)
2218, 21anim12i 614 . . . . . . . . . . . 12 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝑧𝑉𝑧𝑦))
23 eldifsn 4752 . . . . . . . . . . . 12 (𝑧 ∈ (𝑉 ∖ {𝑦}) ↔ (𝑧𝑉𝑧𝑦))
2422, 23sylibr 233 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → 𝑧 ∈ (𝑉 ∖ {𝑦}))
25 preq1 4699 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛, 𝑦} = {𝑧, 𝑦})
2625eleq1d 2823 . . . . . . . . . . . 12 (𝑛 = 𝑧 → ({𝑛, 𝑦} ∈ 𝐸 ↔ {𝑧, 𝑦} ∈ 𝐸))
2726rspcv 3580 . . . . . . . . . . 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 4698 . . . . . . . . . . . . . . . 16 {𝑦, 𝑧} = {𝑧, 𝑦}
3129, 30eqtr2di 2794 . . . . . . . . . . . . . . 15 (𝑝 = {𝑦, 𝑧} → {𝑧, 𝑦} = 𝑝)
3231eleq1d 2823 . . . . . . . . . . . . . 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 728 . . . . . . . . . . 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 414 . . . . . . . 8 ((𝑦𝑉𝑧𝑉) → ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸))))
3938rexlimivv 3197 . . . . . . 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 408 . . . . 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 3955 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ 𝐸)
448, 43eqssd 3966 . 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 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  {crab 3410  cdif 3912  wss 3915  𝒫 cpw 4565  {csn 4591  {cpr 4593  cfv 6501  2c2 12215  chash 14237  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142  ComplUSGraphccusgr 28400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-hash 14238  df-edg 28041  df-upgr 28075  df-umgr 28076  df-usgr 28144  df-nbgr 28323  df-uvtx 28376  df-cplgr 28401  df-cusgr 28402
This theorem is referenced by:  cusgrfilem1  28445
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