MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgredg Structured version   Visualization version   GIF version

Theorem cusgredg 26721
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 26720 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸))
4 usgredgss 26457 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
51pweqi 4381 . . . . . 6 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
6 rabeq 3404 . . . . . 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 3870 . . . 4 (𝐺 ∈ USGraph → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
98adantr 474 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
10 elss2prb 13557 . . . . 5 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ↔ ∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}))
11 sneq 4406 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑣} = {𝑦})
1211difeq2d 3954 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑦}))
13 preq2 4486 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑛, 𝑣} = {𝑛, 𝑦})
1413eleq1d 2890 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ({𝑛, 𝑣} ∈ 𝐸 ↔ {𝑛, 𝑦} ∈ 𝐸))
1512, 14raleqbidv 3363 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (∀𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1615rspcv 3521 . . . . . . . . . . . 12 (𝑦𝑉 → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1716adantr 474 . . . . . . . . . . 11 ((𝑦𝑉𝑧𝑉) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1817adantr 474 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
19 simpr 479 . . . . . . . . . . . . 13 ((𝑦𝑉𝑧𝑉) → 𝑧𝑉)
20 necom 3051 . . . . . . . . . . . . . . 15 (𝑦𝑧𝑧𝑦)
2120biimpi 208 . . . . . . . . . . . . . 14 (𝑦𝑧𝑧𝑦)
2221adantr 474 . . . . . . . . . . . . 13 ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑧𝑦)
2319, 22anim12i 608 . . . . . . . . . . . 12 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝑧𝑉𝑧𝑦))
24 eldifsn 4535 . . . . . . . . . . . 12 (𝑧 ∈ (𝑉 ∖ {𝑦}) ↔ (𝑧𝑉𝑧𝑦))
2523, 24sylibr 226 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → 𝑧 ∈ (𝑉 ∖ {𝑦}))
26 preq1 4485 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛, 𝑦} = {𝑧, 𝑦})
2726eleq1d 2890 . . . . . . . . . . . 12 (𝑛 = 𝑧 → ({𝑛, 𝑦} ∈ 𝐸 ↔ {𝑧, 𝑦} ∈ 𝐸))
2827rspcv 3521 . . . . . . . . . . 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 4484 . . . . . . . . . . . . . . . 16 {𝑦, 𝑧} = {𝑧, 𝑦}
3230, 31syl6req 2877 . . . . . . . . . . . . . . 15 (𝑝 = {𝑦, 𝑧} → {𝑧, 𝑦} = 𝑝)
3332eleq1d 2890 . . . . . . . . . . . . . 14 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3433biimpd 221 . . . . . . . . . . . . 13 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3534a1d 25 . . . . . . . . . . . 12 (𝑝 = {𝑦, 𝑧} → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3635ad2antll 722 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3736com23 86 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → ({𝑧, 𝑦} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3818, 29, 373syld 60 . . . . . . . . 9 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
3938ex 403 . . . . . . . 8 ((𝑦𝑉𝑧𝑉) → ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸))))
4039rexlimivv 3245 . . . . . . 7 (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸)))
4140com13 88 . . . . . 6 (𝐺 ∈ USGraph → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸)))
4241imp 397 . . . . 5 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸))
4310, 42syl5bi 234 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → 𝑝𝐸))
4443ssrdv 3832 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ⊆ 𝐸)
459, 44eqssd 3843 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
463, 45sylbi 209 1 (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wne 2998  wral 3116  wrex 3117  {crab 3120  cdif 3794  wss 3797  𝒫 cpw 4377  {csn 4396  {cpr 4398  cfv 6122  2c2 11405  chash 13409  Vtxcvtx 26293  Edgcedg 26344  USGraphcusgr 26447  ComplUSGraphccusgr 26707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-2o 7826  df-oadd 7829  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-card 9077  df-cda 9304  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-nn 11350  df-2 11413  df-n0 11618  df-xnn0 11690  df-z 11704  df-uz 11968  df-fz 12619  df-hash 13410  df-edg 26345  df-upgr 26379  df-umgr 26380  df-usgr 26449  df-nbgr 26629  df-uvtx 26683  df-cplgr 26708  df-cusgr 26709
This theorem is referenced by:  cusgrfilem1  26752
  Copyright terms: Public domain W3C validator