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Theorem frgrwopreglem4a 27687
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 27692 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 6437 . . . . . 6 (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌))
21a1i 11 . . . . 5 ((𝑋𝑉𝑌𝑉) → (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌)))
32necon3d 3020 . . . 4 ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → 𝑋𝑌))
43imp 397 . . 3 (((𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
543adant1 1164 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
6 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
7 frgrncvvdeq.d . . . . . . 7 𝐷 = (VtxDeg‘𝐺)
86, 7frgrncvvdeq 27686 . . . . . 6 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
9 oveq2 6918 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3108 . . . . . . . . . . 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 6446 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑦)))
1311, 12imbi12d 336 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦))))
14 neleq1 3107 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6437 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝐷𝑦) = (𝐷𝑌))
1615eqeq2d 2835 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝐷𝑋) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑌)))
1714, 16imbi12d 336 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
18 simpll 783 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑋𝑉)
19 sneq 4409 . . . . . . . . . . 11 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2019difeq2d 3957 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
2120adantl 475 . . . . . . . . 9 ((((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22 simpr 479 . . . . . . . . . . 11 ((𝑋𝑉𝑌𝑉) → 𝑌𝑉)
23 necom 3052 . . . . . . . . . . . 12 (𝑋𝑌𝑌𝑋)
2423biimpi 208 . . . . . . . . . . 11 (𝑋𝑌𝑌𝑋)
2522, 24anim12i 606 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝑌𝑉𝑌𝑋))
26 eldifsn 4538 . . . . . . . . . 10 (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌𝑉𝑌𝑋))
2725, 26sylibr 226 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
2813, 17, 18, 21, 27rspc2vd 27642 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
29 nnel 3111 . . . . . . . . . . 11 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 26667 . . . . . . . . . . . . . . . 16 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31 frgrusgr 27637 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edg‘𝐺)
3332nbusgreledg 26657 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3534biimpd 221 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
3630, 35syl5bi 234 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
3736imp 397 . . . . . . . . . . . . . 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 404 . . . . . . . . . . . 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 209 . . . . . . . . . 10 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
42 eqneqall 3010 . . . . . . . . . . 11 ((𝐷𝑋) = (𝐷𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
43422a1d 26 . . . . . . . . . 10 ((𝐷𝑋) = (𝐷𝑌) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4441, 43ja 175 . . . . . . . . 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 406 . . . 4 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → (𝑋𝑌 → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
5049com34 91 . . 3 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
51503imp 1141 . 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 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wnel 3102  wral 3117  cdif 3795  {csn 4399  {cpr 4401  cfv 6127  (class class class)co 6910  Vtxcvtx 26301  Edgcedg 26352  USGraphcusgr 26455   NeighbVtx cnbgr 26636  VtxDegcvtxdg 26770   FriendGraph cfrgr 27633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-xadd 12240  df-fz 12627  df-hash 13418  df-edg 26353  df-uhgr 26363  df-ushgr 26364  df-upgr 26387  df-umgr 26388  df-uspgr 26456  df-usgr 26457  df-nbgr 26637  df-vtxdg 26771  df-frgr 27634
This theorem is referenced by:  frgrwopreglem5a  27688  frgrwopreglem4  27692
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