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Theorem frgrwopreglem4 41486
Description: Lemma 4 for frgrwopreg 41488. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
frgrwopreg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem4 (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝐺,𝑎,𝑏,𝑥
Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑥,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑎,𝑏)   𝐾(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.v . . . 4 𝑉 = (Vtx‘𝐺)
2 frgrwopreg.d . . . 4 𝐷 = (VtxDeg‘𝐺)
3 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
4 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
51, 2, 3, 4frgrwopreglem3 41485 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
61, 2frgrncvvdeq 41482 . . . 4 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
7 elrabi 3327 . . . . . . . . 9 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
87, 3eleq2s 2705 . . . . . . . 8 (𝑎𝐴𝑎𝑉)
9 sneq 4134 . . . . . . . . . . 11 (𝑥 = 𝑎 → {𝑥} = {𝑎})
109difeq2d 3689 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑎}))
11 oveq2 6535 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎))
12 neleq2 2888 . . . . . . . . . . . 12 ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
1311, 12syl 17 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
14 fveq2 6088 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
1514eqeq1d 2611 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑎) = (𝐷𝑦)))
1613, 15imbi12d 332 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
1710, 16raleqbidv 3128 . . . . . . . . 9 (𝑥 = 𝑎 → (∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
1817rspcv 3277 . . . . . . . 8 (𝑎𝑉 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
198, 18syl 17 . . . . . . 7 (𝑎𝐴 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
2019adantr 479 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
214eleq2i 2679 . . . . . . . . . 10 (𝑏𝐵𝑏 ∈ (𝑉𝐴))
22 eldif 3549 . . . . . . . . . 10 (𝑏 ∈ (𝑉𝐴) ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
2321, 22bitri 262 . . . . . . . . 9 (𝑏𝐵 ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
24 simpll 785 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏𝑉)
25 eleq1a 2682 . . . . . . . . . . . . . . 15 (𝑎𝐴 → (𝑏 = 𝑎𝑏𝐴))
2625con3rr3 149 . . . . . . . . . . . . . 14 𝑏𝐴 → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2726adantl 480 . . . . . . . . . . . . 13 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2827imp 443 . . . . . . . . . . . 12 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 = 𝑎)
29 velsn 4140 . . . . . . . . . . . 12 (𝑏 ∈ {𝑎} ↔ 𝑏 = 𝑎)
3028, 29sylnibr 317 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 ∈ {𝑎})
3124, 30eldifd 3550 . . . . . . . . . 10 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
3231ex 448 . . . . . . . . 9 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3323, 32sylbi 205 . . . . . . . 8 (𝑏𝐵 → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3433impcom 444 . . . . . . 7 ((𝑎𝐴𝑏𝐵) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
35 neleq1 2887 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∉ (𝐺 NeighbVtx 𝑎)))
36 fveq2 6088 . . . . . . . . . 10 (𝑦 = 𝑏 → (𝐷𝑦) = (𝐷𝑏))
3736eqeq2d 2619 . . . . . . . . 9 (𝑦 = 𝑏 → ((𝐷𝑎) = (𝐷𝑦) ↔ (𝐷𝑎) = (𝐷𝑏)))
3835, 37imbi12d 332 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) ↔ (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
3938rspcv 3277 . . . . . . 7 (𝑏 ∈ (𝑉 ∖ {𝑎}) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
4034, 39syl 17 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
41 nnel 2891 . . . . . . . . 9 𝑏 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
42 frgrusgr 41434 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
43 frgrwopreg.e . . . . . . . . . . . . . . . 16 𝐸 = (Edg‘𝐺)
4443nbusgreledg 40577 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
4542, 44syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
46 prcom 4210 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
4746eleq1i 2678 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)
4845, 47syl6bb 274 . . . . . . . . . . . . 13 (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑎, 𝑏} ∈ 𝐸))
4948biimpa 499 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → {𝑎, 𝑏} ∈ 𝐸)
5049a1d 25 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))
5150expcom 449 . . . . . . . . . 10 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
5251a1d 25 . . . . . . . . 9 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5341, 52sylbi 205 . . . . . . . 8 𝑏 ∉ (𝐺 NeighbVtx 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
54 eqneqall 2792 . . . . . . . . 9 ((𝐷𝑎) = (𝐷𝑏) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))
55542a1d 26 . . . . . . . 8 ((𝐷𝑎) = (𝐷𝑏) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5653, 55ja 171 . . . . . . 7 ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏)) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5756com12 32 . . . . . 6 ((𝑎𝐴𝑏𝐵) → ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏)) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5820, 40, 573syld 57 . . . . 5 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5958com3l 86 . . . 4 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
606, 59mpcom 37 . . 3 (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
615, 60mpdi 43 . 2 (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → {𝑎, 𝑏} ∈ 𝐸))
6261ralrimivv 2952 1 (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wnel 2780  wral 2895  {crab 2899  cdif 3536  {csn 4124  {cpr 4126  cfv 5790  (class class class)co 6527  Vtxcvtx 40231  Edgcedga 40353   USGraph cusgr 40381   NeighbVtx cnbgr 40552  VtxDegcvtxdg 40683   FriendGraph cfrgr 41430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-xadd 11779  df-fz 12153  df-hash 12935  df-xnn0 40200  df-uhgr 40282  df-ushgr 40283  df-upgr 40310  df-umgr 40311  df-edga 40354  df-uspgr 40382  df-usgr 40383  df-nbgr 40556  df-vtxdg 40684  df-frgr 41431
This theorem is referenced by:  frgrwopreglem5  41487  frgrwopreg1  41489  frgrwopreg2  41490
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