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Theorem frgrwopreglem4a 29831
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 29836 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 2960 . . . 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 1129 . 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 29830 . . . . . 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 7420 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3052 . . . . . . . . . . 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 3051 . . . . . . . . . 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 2742 . . . . . . . . . 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 764 . . . . . . . . 9 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ 𝑋 ∈ 𝑉)
19 sneq 4638 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2019difeq2d 4122 . . . . . . . . . 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 2993 . . . . . . . . . . . 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 4790 . . . . . . . . . 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 3944 . . . . . . . 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 3055 . . . . . . . . . . 11 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 28888 . . . . . . . . . . . . . . . 16 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
31 frgrusgr 29782 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edgβ€˜πΊ)
3332nbusgreledg 28878 . . . . . . . . . . . . . . . . . 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 2950 . . . . . . . . . . 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 1110 . 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 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   βˆ‰ wnel 3045  βˆ€wral 3060   βˆ– cdif 3945  {csn 4628  {cpr 4630  β€˜cfv 6543  (class class class)co 7412  Vtxcvtx 28524  Edgcedg 28575  USGraphcusgr 28677   NeighbVtx cnbgr 28857  VtxDegcvtxdg 28990   FriendGraph cfrgr 29779
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 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
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 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-xadd 13098  df-fz 13490  df-hash 14296  df-edg 28576  df-uhgr 28586  df-ushgr 28587  df-upgr 28610  df-umgr 28611  df-uspgr 28678  df-usgr 28679  df-nbgr 28858  df-vtxdg 28991  df-frgr 29780
This theorem is referenced by:  frgrwopreglem5a  29832  frgrwopreglem4  29836
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