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Theorem frgrwopreglem4a 29996
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30001 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 6891 . . . . . 6 (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌))
21a1i 11 . . . . 5 ((𝑋𝑉𝑌𝑉) → (𝑋 = 𝑌 → (𝐷𝑋) = (𝐷𝑌)))
32necon3d 2960 . . . 4 ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → 𝑋𝑌))
43imp 406 . . 3 (((𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
543adant1 1129 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋𝑉𝑌𝑉) ∧ (𝐷𝑋) ≠ (𝐷𝑌)) → 𝑋𝑌)
6 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
7 frgrncvvdeq.d . . . . . . 7 𝐷 = (VtxDeg‘𝐺)
86, 7frgrncvvdeq 29995 . . . . . 6 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
9 oveq2 7420 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3052 . . . . . . . . . . 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 6900 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑦)))
1311, 12imbi12d 344 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦))))
14 neleq1 3051 . . . . . . . . . 10 (𝑦 = 𝑌 → (𝑦 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∉ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6891 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝐷𝑦) = (𝐷𝑌))
1615eqeq2d 2742 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝐷𝑋) = (𝐷𝑦) ↔ (𝐷𝑋) = (𝐷𝑌)))
1714, 16imbi12d 344 . . . . . . . . 9 (𝑦 = 𝑌 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑦)) ↔ (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
18 simpll 764 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑋𝑉)
19 sneq 4638 . . . . . . . . . . 11 (𝑥 = 𝑋 → {𝑥} = {𝑋})
2019difeq2d 4122 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
2120adantl 481 . . . . . . . . 9 ((((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) ∧ 𝑥 = 𝑋) → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑋}))
22 simpr 484 . . . . . . . . . . 11 ((𝑋𝑉𝑌𝑉) → 𝑌𝑉)
23 necom 2993 . . . . . . . . . . . 12 (𝑋𝑌𝑌𝑋)
2423biimpi 215 . . . . . . . . . . 11 (𝑋𝑌𝑌𝑋)
2522, 24anim12i 612 . . . . . . . . . 10 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝑌𝑉𝑌𝑋))
26 eldifsn 4790 . . . . . . . . . 10 (𝑌 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑌𝑉𝑌𝑋))
2725, 26sylibr 233 . . . . . . . . 9 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → 𝑌 ∈ (𝑉 ∖ {𝑋}))
2813, 17, 18, 21, 27rspc2vd 3944 . . . . . . . 8 (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (𝐷𝑋) = (𝐷𝑌))))
29 nnel 3055 . . . . . . . . . . 11 𝑌 ∉ (𝐺 NeighbVtx 𝑋) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 29053 . . . . . . . . . . . . . . . 16 (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
31 frgrusgr 29947 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edg‘𝐺)
3332nbusgreledg 29043 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ 𝐸))
3534biimpd 228 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ 𝐸))
3630, 35biimtrid 241 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ 𝐸))
3736imp 406 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 216 . . . . . . . . . 10 𝑌 ∉ (𝐺 NeighbVtx 𝑋) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
42 eqneqall 2950 . . . . . . . . . . 11 ((𝐷𝑋) = (𝐷𝑌) → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))
43422a1d 26 . . . . . . . . . 10 ((𝐷𝑋) = (𝐷𝑌) → (((𝑋𝑉𝑌𝑉) ∧ 𝑋𝑌) → (𝐺 ∈ FriendGraph → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
4441, 43ja 186 . . . . . . . . 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 415 . . . 4 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → (𝑋𝑌 → ((𝐷𝑋) ≠ (𝐷𝑌) → {𝑋, 𝑌} ∈ 𝐸))))
5049com34 91 . . 3 (𝐺 ∈ FriendGraph → ((𝑋𝑉𝑌𝑉) → ((𝐷𝑋) ≠ (𝐷𝑌) → (𝑋𝑌 → {𝑋, 𝑌} ∈ 𝐸))))
51503imp 1110 . 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wnel 3045  wral 3060  cdif 3945  {csn 4628  {cpr 4630  cfv 6543  (class class class)co 7412  Vtxcvtx 28689  Edgcedg 28740  USGraphcusgr 28842   NeighbVtx cnbgr 29022  VtxDegcvtxdg 29155   FriendGraph cfrgr 29944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-xadd 13100  df-fz 13492  df-hash 14298  df-edg 28741  df-uhgr 28751  df-ushgr 28752  df-upgr 28775  df-umgr 28776  df-uspgr 28843  df-usgr 28844  df-nbgr 29023  df-vtxdg 29156  df-frgr 29945
This theorem is referenced by:  frgrwopreglem5a  29997  frgrwopreglem4  30001
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