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Theorem frgrwopreglem4a 27456
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 27461 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 6344 . . . . . 6 (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌))
21a1i 11 . . . . 5 ((𝑋𝑉𝑌𝑉) → (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌)))
32necon3d 2945 . . . 4 ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → 𝑋𝑌))
43imp 444 . . 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 27455 . . . . . 6 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
9 oveq2 6813 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3033 . . . . . . . . . . 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 fveq2 6344 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
1312eqeq1d 2754 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑦)))
1411, 13imbi12d 333 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦))))
15 neleq1 3032 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
16 fveq2 6344 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝐷𝑦) = (𝐷𝑌))
1716eqeq2d 2762 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝐷𝑋) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑌)))
1815, 17imbi12d 333 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
19 simpll 807 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑋𝑉)
20 sneq 4323 . . . . . . . . . . 11 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2120difeq2d 3863 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
2221adantl 473 . . . . . . . . 9 ((((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
23 simpr 479 . . . . . . . . . . 11 ((𝑋𝑉𝑌𝑉) → 𝑌𝑉)
24 necom 2977 . . . . . . . . . . . 12 (𝑋𝑌𝑌𝑋)
2524biimpi 206 . . . . . . . . . . 11 (𝑋𝑌𝑌𝑋)
2623, 25anim12i 591 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝑌𝑉𝑌𝑋))
27 eldifsn 4454 . . . . . . . . . 10 (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌𝑉𝑌𝑋))
2826, 27sylibr 224 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
2914, 18, 19, 22, 28rspc2vd 27411 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
30 nnel 3036 . . . . . . . . . . 11 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
31 nbgrsym 26454 . . . . . . . . . . . . . . . 16 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
32 frgrusgr 27406 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
33 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edg‘𝐺)
3433nbusgreledg 26440 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3532, 34syl 17 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3635biimpd 219 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
3731, 36syl5bi 232 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
3837imp 444 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ 𝐸)
3938a1d 25 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
4039expcom 450 . . . . . . . . . . . 12 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
4140a1d 25 . . . . . . . . . . 11 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4230, 41sylbi 207 . . . . . . . . . 10 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
43 eqneqall 2935 . . . . . . . . . . 11 ((𝐷𝑋) = (𝐷𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
44432a1d 26 . . . . . . . . . 10 ((𝐷𝑋) = (𝐷𝑌) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4542, 44ja 173 . . . . . . . . 9 ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌)) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4645com12 32 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌)) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4729, 46syld 47 . . . . . . 7 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4847com3l 89 . . . . . 6 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
498, 48mpcom 38 . . . . 5 (𝐺 ∈ FriendGraph → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸)))
5049expd 451 . . . 4 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → (𝑋𝑌 → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
5150com34 91 . . 3 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
52513imp 1101 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))
535, 52mpd 15 1 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → {𝑋, 𝑌} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wnel 3027  wral 3042  cdif 3704  {csn 4313  {cpr 4315  cfv 6041  (class class class)co 6805  Vtxcvtx 26065  Edgcedg 26130  USGraphcusgr 26235   NeighbVtx cnbgr 26415  VtxDegcvtxdg 26563   FriendGraph cfrgr 27402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-xnn0 11548  df-z 11562  df-uz 11872  df-xadd 12132  df-fz 12512  df-hash 13304  df-edg 26131  df-uhgr 26144  df-ushgr 26145  df-upgr 26168  df-umgr 26169  df-uspgr 26236  df-usgr 26237  df-nbgr 26416  df-vtxdg 26564  df-frgr 27403
This theorem is referenced by:  frgrwopreglem5a  27457  frgrwopreglem4  27461
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