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Theorem frgrwopreglem4a 29552
Description: In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 29557 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 6888 . . . . . 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 2961 . . . 4 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π·β€˜π‘‹) β‰  (π·β€˜π‘Œ) β†’ 𝑋 β‰  π‘Œ))
43imp 407 . . 3 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ (π·β€˜π‘‹) β‰  (π·β€˜π‘Œ)) β†’ 𝑋 β‰  π‘Œ)
543adant1 1130 . 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 29551 . . . . . 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 7413 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
10 neleq2 3053 . . . . . . . . . . 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 6897 . . . . . . . . . 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 3052 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑦 βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋)))
15 fveq2 6888 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (π·β€˜π‘¦) = (π·β€˜π‘Œ))
1615eqeq2d 2743 . . . . . . . . . 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 765 . . . . . . . . 9 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ 𝑋 ∈ 𝑉)
19 sneq 4637 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
2019difeq2d 4121 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
2120adantl 482 . . . . . . . . 9 ((((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) ∧ π‘₯ = 𝑋) β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {𝑋}))
22 simpr 485 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ 𝑉)
23 necom 2994 . . . . . . . . . . . 12 (𝑋 β‰  π‘Œ ↔ π‘Œ β‰  𝑋)
2423biimpi 215 . . . . . . . . . . 11 (𝑋 β‰  π‘Œ β†’ π‘Œ β‰  𝑋)
2522, 24anim12i 613 . . . . . . . . . 10 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑋 β‰  π‘Œ) β†’ (π‘Œ ∈ 𝑉 ∧ π‘Œ β‰  𝑋))
26 eldifsn 4789 . . . . . . . . . 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 3943 . . . . . . . 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 3056 . . . . . . . . . . 11 (Β¬ π‘Œ βˆ‰ (𝐺 NeighbVtx 𝑋) ↔ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))
30 nbgrsym 28609 . . . . . . . . . . . . . . . 16 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
31 frgrusgr 29503 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ FriendGraph β†’ 𝐺 ∈ USGraph)
32 frgrwopreglem4a.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (Edgβ€˜πΊ)
3332nbusgreledg 28599 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 2951 . . . . . . . . . . 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 416 . . . 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 1111 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ‰ wnel 3046  βˆ€wral 3061   βˆ– cdif 3944  {csn 4627  {cpr 4629  β€˜cfv 6540  (class class class)co 7405  Vtxcvtx 28245  Edgcedg 28296  USGraphcusgr 28398   NeighbVtx cnbgr 28578  VtxDegcvtxdg 28711   FriendGraph cfrgr 29500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-xadd 13089  df-fz 13481  df-hash 14287  df-edg 28297  df-uhgr 28307  df-ushgr 28308  df-upgr 28331  df-umgr 28332  df-uspgr 28399  df-usgr 28400  df-nbgr 28579  df-vtxdg 28712  df-frgr 29501
This theorem is referenced by:  frgrwopreglem5a  29553  frgrwopreglem4  29557
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