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Theorem frgrwopreglem4a 30176
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 30181 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 6894 . . . . . 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 2951 . . . 4 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ 𝑋 β‰  π‘Œ))
43imp 405 . . 3 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ 𝑋 β‰  π‘Œ)
543adant1 1127 . 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 30175 . . . . . 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 7425 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3043 . . . . . . . . . . 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 6903 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ ((π·β€˜π‘₯) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘¦)))
1311, 12imbi12d 343 . . . . . . . . 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 3042 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6894 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (π·β€˜π‘¦) = (π·β€˜π‘Œ))
1615eqeq2d 2736 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ ((π·β€˜π‘‹) = (π·β€˜π‘¦) ↔ (π·β€˜π‘‹) = (π·β€˜π‘Œ)))
1714, 16imbi12d 343 . . . . . . . . 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 765 . . . . . . . . 9 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ 𝑋 ∈ 𝑉)
19 sneq 4639 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2019difeq2d 4119 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
2120adantl 480 . . . . . . . . 9 ((((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) ∧ π‘₯ = 𝑋) β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
22 simpr 483 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ 𝑉)
23 necom 2984 . . . . . . . . . . . 12 (𝑋 β‰  π‘Œ ↔ π‘Œ β‰  𝑋)
2423biimpi 215 . . . . . . . . . . 11 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2522, 24anim12i 611 . . . . . . . . . 10 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝑉 ∧ π‘Œ β‰  𝑋))
26 eldifsn 4791 . . . . . . . . . 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 3941 . . . . . . . 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 3046 . . . . . . . . . . 11 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 29232 . . . . . . . . . . . . . . . 16 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
31 frgrusgr 30127 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edgβ€˜πΊ)
3332nbusgreledg 29222 . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 2941 . . . . . . . . . . 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 414 . . . 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 1108 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   βˆ‰ wnel 3036  βˆ€wral 3051   βˆ– cdif 3942  {csn 4629  {cpr 4631  β€˜cfv 6547  (class class class)co 7417  Vtxcvtx 28865  Edgcedg 28916  USGraphcusgr 29018   NeighbVtx cnbgr 29201  VtxDegcvtxdg 29335   FriendGraph cfrgr 30124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-xnn0 12575  df-z 12589  df-uz 12853  df-xadd 13125  df-fz 13517  df-hash 14322  df-edg 28917  df-uhgr 28927  df-ushgr 28928  df-upgr 28951  df-umgr 28952  df-uspgr 29019  df-usgr 29020  df-nbgr 29202  df-vtxdg 29336  df-frgr 30125
This theorem is referenced by:  frgrwopreglem5a  30177  frgrwopreglem4  30181
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