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Theorem nb3gr2nb 26330
 Description: If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
Assertion
Ref Expression
nb3gr2nb (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))

Proof of Theorem nb3gr2nb
StepHypRef Expression
1 prcom 4299 . . . . . . . . 9 {𝐴, 𝐶} = {𝐶, 𝐴}
21eleq1i 2721 . . . . . . . 8 ({𝐴, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐴} ∈ (Edg‘𝐺))
32biimpi 206 . . . . . . 7 ({𝐴, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
43adantl 481 . . . . . 6 (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐴} ∈ (Edg‘𝐺))
5 prcom 4299 . . . . . . . . 9 {𝐵, 𝐶} = {𝐶, 𝐵}
65eleq1i 2721 . . . . . . . 8 ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ {𝐶, 𝐵} ∈ (Edg‘𝐺))
76biimpi 206 . . . . . . 7 ({𝐵, 𝐶} ∈ (Edg‘𝐺) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
87adantl 481 . . . . . 6 (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺)) → {𝐶, 𝐵} ∈ (Edg‘𝐺))
94, 8anim12i 589 . . . . 5 ((({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))) → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺)))
109a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))) → ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
11 eqid 2651 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
12 eqid 2651 . . . . . 6 (Edg‘𝐺) = (Edg‘𝐺)
13 simprr 811 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
14 simprl 809 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶})
15 simpl 472 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐴𝑋𝐵𝑌𝐶𝑍))
1611, 12, 13, 14, 15nb3grprlem1 26326 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺))))
17 3ancoma 1062 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐴𝑋𝐶𝑍))
1817biimpi 206 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐵𝑌𝐴𝑋𝐶𝑍))
19 tpcoma 4317 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
2019eqeq2i 2663 . . . . . . . 8 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
2120biimpi 206 . . . . . . 7 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
2221anim1i 591 . . . . . 6 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph))
23 simprr 811 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
24 simprl 809 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐵, 𝐴, 𝐶})
25 simpl 472 . . . . . . 7 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → (𝐵𝑌𝐴𝑋𝐶𝑍))
2611, 12, 23, 24, 25nb3grprlem1 26326 . . . . . 6 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐵, 𝐴, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
2718, 22, 26syl2an 493 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺))))
2816, 27anbi12d 747 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ (({𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐶} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐶} ∈ (Edg‘𝐺)))))
29 3anrot 1060 . . . . . 6 ((𝐶𝑍𝐴𝑋𝐵𝑌) ↔ (𝐴𝑋𝐵𝑌𝐶𝑍))
3029biimpri 218 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐴𝑋𝐵𝑌))
31 tprot 4316 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
3231eqcomi 2660 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
3332eqeq2i 2663 . . . . . . 7 ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ↔ (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
3433anbi1i 731 . . . . . 6 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ↔ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
3534biimpi 206 . . . . 5 (((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph))
36 simprr 811 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → 𝐺 ∈ USGraph)
37 simprl 809 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (Vtx‘𝐺) = {𝐶, 𝐴, 𝐵})
38 simpl 472 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → (𝐶𝑍𝐴𝑋𝐵𝑌))
3911, 12, 36, 37, 38nb3grprlem1 26326 . . . . 5 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ ((Vtx‘𝐺) = {𝐶, 𝐴, 𝐵} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
4030, 35, 39syl2an 493 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐶, 𝐵} ∈ (Edg‘𝐺))))
4110, 28, 403imtr4d 283 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) → (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}))
4241pm4.71d 667 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
43 df-3an 1056 . 2 (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}) ↔ (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵}))
4442, 43syl6bbr 278 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cpr 4212  {ctp 4214  ‘cfv 5926  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-umgr 26023  df-usgr 26091  df-nbgr 26270 This theorem is referenced by: (None)
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