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Theorem frgrwopreglem4a 29296
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 29301 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 6847 . . . . . 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 2965 . . . 4 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ 𝑋 β‰  π‘Œ))
43imp 408 . . 3 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ 𝑋 β‰  π‘Œ)
543adant1 1131 . 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 29295 . . . . . 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 7370 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3056 . . . . . . . . . . 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 6856 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘₯) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘¦)))
1311, 12imbi12d 345 . . . . . . . . 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 3055 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6847 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (π·β€˜π‘¦) = (π·β€˜π‘Œ))
1615eqeq2d 2748 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ ((π·β€˜π‘‹) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘Œ)))
1714, 16imbi12d 345 . . . . . . . . 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 4601 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2019difeq2d 4087 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
2120adantl 483 . . . . . . . . 9 ((((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) ∧ π‘₯ = 𝑋) β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
22 simpr 486 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ 𝑉)
23 necom 2998 . . . . . . . . . . . 12 (𝑋 β‰  π‘Œ ↔ π‘Œ β‰  𝑋)
2423biimpi 215 . . . . . . . . . . 11 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2522, 24anim12i 614 . . . . . . . . . 10 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝑉 ∧ π‘Œ β‰  𝑋))
26 eldifsn 4752 . . . . . . . . . 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 3911 . . . . . . . 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 3059 . . . . . . . . . . 11 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 28353 . . . . . . . . . . . . . . . 16 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
31 frgrusgr 29247 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edgβ€˜πΊ)
3332nbusgreledg 28343 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 2955 . . . . . . . . . . 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 417 . . . 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 1112 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ‰ wnel 3050  βˆ€wral 3065   βˆ– cdif 3912  {csn 4591  {cpr 4593  β€˜cfv 6501  (class class class)co 7362  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142   NeighbVtx cnbgr 28322  VtxDegcvtxdg 28455   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-xadd 13041  df-fz 13432  df-hash 14238  df-edg 28041  df-uhgr 28051  df-ushgr 28052  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-nbgr 28323  df-vtxdg 28456  df-frgr 29245
This theorem is referenced by:  frgrwopreglem5a  29297  frgrwopreglem4  29301
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