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

Theorem cusgredg 26314
 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 26313 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸))
4 usgredgss 26048 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
51pweqi 4160 . . . . . 6 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
6 rabeq 3190 . . . . . 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 3644 . . . 4 (𝐺 ∈ USGraph → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
98adantr 481 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
10 elss2prb 13264 . . . . 5 (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ ∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}))
11 sneq 4185 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑣} = {𝑦})
1211difeq2d 3726 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑦}))
13 preq2 4267 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → {𝑛, 𝑣} = {𝑛, 𝑦})
1413eleq1d 2685 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → ({𝑛, 𝑣} ∈ 𝐸 ↔ {𝑛, 𝑦} ∈ 𝐸))
1512, 14raleqbidv 3150 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (∀𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1615rspcv 3303 . . . . . . . . . . . 12 (𝑦𝑉 → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1716adantr 481 . . . . . . . . . . 11 ((𝑦𝑉𝑧𝑉) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
1817adantr 481 . . . . . . . . . 10 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → ∀𝑛 ∈ (𝑉 ∖ {𝑦}){𝑛, 𝑦} ∈ 𝐸))
19 simpr 477 . . . . . . . . . . . . 13 ((𝑦𝑉𝑧𝑉) → 𝑧𝑉)
20 necom 2846 . . . . . . . . . . . . . . 15 (𝑦𝑧𝑧𝑦)
2120biimpi 206 . . . . . . . . . . . . . 14 (𝑦𝑧𝑧𝑦)
2221adantr 481 . . . . . . . . . . . . 13 ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑧𝑦)
2319, 22anim12i 590 . . . . . . . . . . . 12 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → (𝑧𝑉𝑧𝑦))
24 eldifsn 4315 . . . . . . . . . . . 12 (𝑧 ∈ (𝑉 ∖ {𝑦}) ↔ (𝑧𝑉𝑧𝑦))
2523, 24sylibr 224 . . . . . . . . . . 11 (((𝑦𝑉𝑧𝑉) ∧ (𝑦𝑧𝑝 = {𝑦, 𝑧})) → 𝑧 ∈ (𝑉 ∖ {𝑦}))
26 preq1 4266 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛, 𝑦} = {𝑧, 𝑦})
2726eleq1d 2685 . . . . . . . . . . . 12 (𝑛 = 𝑧 → ({𝑛, 𝑦} ∈ 𝐸 ↔ {𝑧, 𝑦} ∈ 𝐸))
2827rspcv 3303 . . . . . . . . . . 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 4265 . . . . . . . . . . . . . . . 16 {𝑦, 𝑧} = {𝑧, 𝑦}
3230, 31syl6req 2672 . . . . . . . . . . . . . . 15 (𝑝 = {𝑦, 𝑧} → {𝑧, 𝑦} = 𝑝)
3332eleq1d 2685 . . . . . . . . . . . . . 14 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3433biimpd 219 . . . . . . . . . . . . 13 (𝑝 = {𝑦, 𝑧} → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸))
3534a1d 25 . . . . . . . . . . . 12 (𝑝 = {𝑦, 𝑧} → (𝐺 ∈ USGraph → ({𝑧, 𝑦} ∈ 𝐸𝑝𝐸)))
3635ad2antll 765 . . . . . . . . . . 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 450 . . . . . . . 8 ((𝑦𝑉𝑧𝑉) → ((𝑦𝑧𝑝 = {𝑦, 𝑧}) → (∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑝𝐸))))
4039rexlimivv 3034 . . . . . . 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 445 . . . . 5 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (∃𝑦𝑉𝑧𝑉 (𝑦𝑧𝑝 = {𝑦, 𝑧}) → 𝑝𝐸))
4310, 42syl5bi 232 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → (𝑝 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑝𝐸))
4443ssrdv 3607 . . 3 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⊆ 𝐸)
459, 44eqssd 3618 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣}){𝑛, 𝑣} ∈ 𝐸) → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
463, 45sylbi 207 1 (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  ∃wrex 2912  {crab 2915   ∖ cdif 3569   ⊆ wss 3572  𝒫 cpw 4156  {csn 4175  {cpr 4177  ‘cfv 5886  2c2 11067  #chash 13112  Vtxcvtx 25868  Edgcedg 25933   USGraph cusgr 26038  ComplUSGraphccusgr 26221 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-n0 11290  df-xnn0 11361  df-z 11375  df-uz 11685  df-fz 12324  df-hash 13113  df-edg 25934  df-upgr 25971  df-umgr 25972  df-usgr 26040  df-nbgr 26222  df-uvtxa 26224  df-cplgr 26225  df-cusgr 26226 This theorem is referenced by:  cusgrfilem1  26345
 Copyright terms: Public domain W3C validator