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Theorem frgrwopreglem4a 28004
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 28009 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 4-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreglem4a.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem4a ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → {𝑋, 𝑌} ∈ 𝐸)

Proof of Theorem frgrwopreglem4a
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6666 . . . . . 6 (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌))
21a1i 11 . . . . 5 ((𝑋𝑉𝑌𝑉) → (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌)))
32necon3d 3041 . . . 4 ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → 𝑋𝑌))
43imp 407 . . 3 (((𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
543adant1 1124 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
6 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
7 frgrncvvdeq.d . . . . . . 7 𝐷 = (VtxDeg‘𝐺)
86, 7frgrncvvdeq 28003 . . . . . 6 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
9 oveq2 7159 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3133 . . . . . . . . . . 11 ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
119, 10syl 17 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑋)))
12 fveqeq2 6675 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑦)))
1311, 12imbi12d 346 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦))))
14 neleq1 3132 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6666 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝐷𝑦) = (𝐷𝑌))
1615eqeq2d 2835 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝐷𝑋) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑌)))
1714, 16imbi12d 346 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
18 simpll 763 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑋𝑉)
19 sneq 4573 . . . . . . . . . . 11 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2019difeq2d 4102 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
2120adantl 482 . . . . . . . . 9 ((((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22 simpr 485 . . . . . . . . . . 11 ((𝑋𝑉𝑌𝑉) → 𝑌𝑉)
23 necom 3073 . . . . . . . . . . . 12 (𝑋𝑌𝑌𝑋)
2423biimpi 217 . . . . . . . . . . 11 (𝑋𝑌𝑌𝑋)
2522, 24anim12i 612 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝑌𝑉𝑌𝑋))
26 eldifsn 4717 . . . . . . . . . 10 (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌𝑉𝑌𝑋))
2725, 26sylibr 235 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
2813, 17, 18, 21, 27rspc2vd 3935 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
29 nnel 3136 . . . . . . . . . . 11 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 27060 . . . . . . . . . . . . . . . 16 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31 frgrusgr 27955 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edg‘𝐺)
3332nbusgreledg 27050 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3534biimpd 230 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
3630, 35syl5bi 243 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
3736imp 407 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ 𝐸)
3837a1d 25 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
3938expcom 414 . . . . . . . . . . . 12 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
4039a1d 25 . . . . . . . . . . 11 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4129, 40sylbi 218 . . . . . . . . . 10 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
42 eqneqall 3031 . . . . . . . . . . 11 ((𝐷𝑋) = (𝐷𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
43422a1d 26 . . . . . . . . . 10 ((𝐷𝑋) = (𝐷𝑌) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4441, 43ja 187 . . . . . . . . 9 ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌)) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4544com12 32 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌)) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4628, 45syld 47 . . . . . . 7 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4746com3l 89 . . . . . 6 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
488, 47mpcom 38 . . . . 5 (𝐺 ∈ FriendGraph → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
4948expd 416 . . . 4 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → (𝑋𝑌 → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
5049com34 91 . . 3 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
51503imp 1105 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))
525, 51mpd 15 1 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → {𝑋, 𝑌} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020  wnel 3127  wral 3142  cdif 3936  {csn 4563  {cpr 4565  cfv 6351  (class class class)co 7151  Vtxcvtx 26696  Edgcedg 26747  USGraphcusgr 26849   NeighbVtx cnbgr 27029  VtxDegcvtxdg 27162   FriendGraph cfrgr 27952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-xadd 12501  df-fz 12886  df-hash 13684  df-edg 26748  df-uhgr 26758  df-ushgr 26759  df-upgr 26782  df-umgr 26783  df-uspgr 26850  df-usgr 26851  df-nbgr 27030  df-vtxdg 27163  df-frgr 27953
This theorem is referenced by:  frgrwopreglem5a  28005  frgrwopreglem4  28009
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