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Theorem frgrwopreglem4a 30107
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30112 without a fixed degree and without using the sets 𝐴 and 𝐡. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 4-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v 𝑉 = (Vtxβ€˜πΊ)
frgrncvvdeq.d 𝐷 = (VtxDegβ€˜πΊ)
frgrwopreglem4a.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
frgrwopreglem4a ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ {𝑋, π‘Œ} ∈ 𝐸)
Proof of Theorem frgrwopreglem4a
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . . 6 (𝑋 = π‘Œ β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ))
21a1i 11 . . . . 5 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 = π‘Œ β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ)))
32necon3d 2956 . . . 4 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ 𝑋 β‰  π‘Œ))
43imp 406 . . 3 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ 𝑋 β‰  π‘Œ)
543adant1 1128 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ 𝑋 β‰  π‘Œ)
6 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
7 frgrncvvdeq.d . . . . . . 7 𝐷 = (VtxDegβ€˜πΊ)
86, 7frgrncvvdeq 30106 . . . . . 6 (𝐺 ∈ FriendGraph β†’ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)))
9 oveq2 7422 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3048 . . . . . . . . . . 11 ((𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋) β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) ↔ 𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋)))
119, 10syl 17 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) ↔ 𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋)))
12 fveqeq2 6900 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘₯) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘¦)))
1311, 12imbi12d 344 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ ((𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)) ↔ (𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘¦))))
14 neleq1 3047 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6891 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (π·β€˜π‘¦) = (π·β€˜π‘Œ))
1615eqeq2d 2738 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ ((π·β€˜π‘‹) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘Œ)))
1714, 16imbi12d 344 . . . . . . . . 9 (𝑦 = π‘Œ β†’ ((𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘¦)) ↔ (π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ))))
18 simpll 766 . . . . . . . . 9 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ 𝑋 ∈ 𝑉)
19 sneq 4634 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2019difeq2d 4118 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
2120adantl 481 . . . . . . . . 9 ((((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) ∧ π‘₯ = 𝑋) β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
22 simpr 484 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ 𝑉)
23 necom 2989 . . . . . . . . . . . 12 (𝑋 β‰  π‘Œ ↔ π‘Œ β‰  𝑋)
2423biimpi 215 . . . . . . . . . . 11 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2522, 24anim12i 612 . . . . . . . . . 10 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝑉 ∧ π‘Œ β‰  𝑋))
26 eldifsn 4786 . . . . . . . . . 10 (π‘Œ ∈ (𝑉 βˆ– {𝑋}) ↔ (π‘Œ ∈ 𝑉 ∧ π‘Œ β‰  𝑋))
2725, 26sylibr 233 . . . . . . . . 9 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ π‘Œ ∈ (𝑉 βˆ– {𝑋}))
2813, 17, 18, 21, 27rspc2vd 3940 . . . . . . . 8 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)) β†’ (π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ))))
29 nnel 3051 . . . . . . . . . . 11 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 29163 . . . . . . . . . . . . . . . 16 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
31 frgrusgr 30058 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edgβ€˜πΊ)
3332nbusgreledg 29153 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph β†’ (𝑋 ∈ (𝐺 NeighbVtx π‘Œ) ↔ {𝑋, π‘Œ} ∈ 𝐸))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FriendGraph β†’ (𝑋 ∈ (𝐺 NeighbVtx π‘Œ) ↔ {𝑋, π‘Œ} ∈ 𝐸))
3534biimpd 228 . . . . . . . . . . . . . . . 16 (𝐺 ∈ FriendGraph β†’ (𝑋 ∈ (𝐺 NeighbVtx π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))
3630, 35biimtrid 241 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ {𝑋, π‘Œ} ∈ 𝐸))
3736imp 406 . . . . . . . . . . . . . 14 ((𝐺 ∈ FriendGraph ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ {𝑋, π‘Œ} ∈ 𝐸)
3837a1d 25 . . . . . . . . . . . . 13 ((𝐺 ∈ FriendGraph ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))
3938expcom 413 . . . . . . . . . . . 12 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸)))
4039a1d 25 . . . . . . . . . . 11 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
4129, 40sylbi 216 . . . . . . . . . 10 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
42 eqneqall 2946 . . . . . . . . . . 11 ((π·β€˜π‘‹) = (π·β€˜π‘Œ) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))
43422a1d 26 . . . . . . . . . 10 ((π·β€˜π‘‹) = (π·β€˜π‘Œ) β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
4441, 43ja 186 . . . . . . . . 9 ((π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ)) β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
4544com12 32 . . . . . . . 8 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ ((π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) β†’ (π·β€˜π‘‹) = (π·β€˜π‘Œ)) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
4628, 45syld 47 . . . . . . 7 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)) β†’ (𝐺 ∈ FriendGraph β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
4746com3l 89 . . . . . 6 (βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ (𝑉 βˆ– {π‘₯})(𝑦 βˆ‰ (𝐺 NeighbVtx π‘₯) β†’ (π·β€˜π‘₯) = (π·β€˜π‘¦)) β†’ (𝐺 ∈ FriendGraph β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
488, 47mpcom 38 . . . . 5 (𝐺 ∈ FriendGraph β†’ (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸)))
4948expd 415 . . . 4 (𝐺 ∈ FriendGraph β†’ ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 β‰  π‘Œ β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ {𝑋, π‘Œ} ∈ 𝐸))))
5049com34 91 . . 3 (𝐺 ∈ FriendGraph β†’ ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ (𝑋 β‰  π‘Œ β†’ {𝑋, π‘Œ} ∈ 𝐸))))
51503imp 1109 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ (𝑋 β‰  π‘Œ β†’ {𝑋, π‘Œ} ∈ 𝐸))
525, 51mpd 15 1 ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ {𝑋, π‘Œ} ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935   βˆ‰ wnel 3041  βˆ€wral 3056   βˆ– cdif 3941  {csn 4624  {cpr 4626  β€˜cfv 6542  (class class class)co 7414  Vtxcvtx 28796  Edgcedg 28847  USGraphcusgr 28949   NeighbVtx cnbgr 29132  VtxDegcvtxdg 29266   FriendGraph cfrgr 30055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-xadd 13117  df-fz 13509  df-hash 14314  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-nbgr 29133  df-vtxdg 29267  df-frgr 30056
This theorem is referenced by:  frgrwopreglem5a  30108  frgrwopreglem4  30112
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