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Theorem cusgredg 40646
 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 40645 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸))
4 usgredgss 40390 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
51pweqi 4112 . . . . . 6 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
6 rabeq 3166 . . . . . 6 (𝒫 𝑉 = 𝒫 (Vtx‘𝐺) → {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
75, 6ax-mp 5 . . . . 5 {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}
84, 2, 73sstr4g 3609 . . . 4 (𝐺 ∈ USGraph → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
98adantr 480 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
10 elss2prb 13123 . . . . 5 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}))
11 sneq 4135 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑣} = {𝑦})
1211difeq2d 3690 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑦}))
13 preq2 4213 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑛, 𝑣} = {𝑛, 𝑦})
1413eleq1d 2672 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ({𝑛, 𝑣} ∈ 𝐸 ↔ {𝑛, 𝑦} ∈ 𝐸))
1512, 14raleqbidv 3129 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (∀𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1615rspcv 3278 . . . . . . . . . . . 12 (𝑦𝑉 → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1716adantr 480 . . . . . . . . . . 11 ((𝑦𝑉𝑧𝑉) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1817adantr 480 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
19 simpr 476 . . . . . . . . . . . . 13 ((𝑦𝑉𝑧𝑉) → 𝑧𝑉)
20 necom 2835 . . . . . . . . . . . . . . 15 (𝑦𝑧𝑧𝑦)
2120biimpi 205 . . . . . . . . . . . . . 14 (𝑦𝑧𝑧𝑦)
2221adantr 480 . . . . . . . . . . . . 13 ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑧𝑦)
2319, 22anim12i 588 . . . . . . . . . . . 12 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝑧𝑉𝑧𝑦))
24 eldifsn 4260 . . . . . . . . . . . 12 (𝑧 ∈ (𝑉 ∖ {𝑦}) ↔ (𝑧𝑉𝑧𝑦))
2523, 24sylibr 223 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → 𝑧 ∈ (𝑉 ∖ {𝑦}))
26 preq1 4212 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛, 𝑦} = {𝑧, 𝑦})
2726eleq1d 2672 . . . . . . . . . . . 12 (𝑛 = 𝑧 → ({𝑛, 𝑦} ∈ 𝐸 ↔ {𝑧, 𝑦} ∈ 𝐸))
2827rspcv 3278 . . . . . . . . . . 11 (𝑧 ∈ (𝑉 ∖ {𝑦}) → (∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸 → {𝑧, 𝑦} ∈ 𝐸))
2925, 28syl 17 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸 → {𝑧, 𝑦} ∈ 𝐸))
30 id 22 . . . . . . . . . . . . . . . 16 (𝑝 = {𝑦, 𝑧} → 𝑝 = {𝑦, 𝑧})
31 prcom 4211 . . . . . . . . . . . . . . . 16 {𝑦, 𝑧} = {𝑧, 𝑦}
3230, 31syl6req 2661 . . . . . . . . . . . . . . 15 (𝑝 = {𝑦, 𝑧} → {𝑧, 𝑦} = 𝑝)
3332eleq1d 2672 . . . . . . . . . . . . . 14 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3433biimpd 218 . . . . . . . . . . . . 13 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3534a1d 25 . . . . . . . . . . . 12 (𝑝 = {𝑦, 𝑧} → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3635ad2antll 761 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3736com23 84 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → ({𝑧, 𝑦} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3818, 29, 373syld 58 . . . . . . . . 9 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3938ex 449 . . . . . . . 8 ((𝑦𝑉𝑧𝑉) → ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸))))
4039rexlimivv 3018 . . . . . . 7 (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
4140com13 86 . . . . . 6 (𝐺 ∈ USGraph → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸)))
4241imp 444 . . . . 5 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸))
4310, 42syl5bi 231 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑝𝐸))
4443ssrdv 3574 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⊆ 𝐸)
459, 44eqssd 3585 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
463, 45sylbi 206 1 (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127  ‘cfv 5804  2c2 10947  #chash 12979  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379  ComplUSGraphccusgr 40553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-uvtxa 40556  df-cplgr 40557  df-cusgr 40558 This theorem is referenced by:  cusgrfilem1  40671
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