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Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrregord013 Structured version   Visualization version   GIF version

Theorem frgrregord013 29339
Description: If a finite friendship graph is 𝐾-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
Hypothesis
Ref Expression
frgrreggt1.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
frgrregord013 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))

Proof of Theorem frgrregord013
Dummy variables 𝑣 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hashcl 14256 . . 3 (𝑉 ∈ Fin β†’ (β™―β€˜π‘‰) ∈ β„•0)
2 ax-1 6 . . . . 5 (((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
3 3ioran 1106 . . . . . 6 (Β¬ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) ↔ (Β¬ (β™―β€˜π‘‰) = 0 ∧ Β¬ (β™―β€˜π‘‰) = 1 ∧ Β¬ (β™―β€˜π‘‰) = 3))
4 df-ne 2944 . . . . . . . . . . . . 13 ((β™―β€˜π‘‰) β‰  0 ↔ Β¬ (β™―β€˜π‘‰) = 0)
5 hasheq0 14263 . . . . . . . . . . . . . . . . . 18 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) = 0 ↔ 𝑉 = βˆ…))
65necon3bid 2988 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) β‰  0 ↔ 𝑉 β‰  βˆ…))
76biimpa 477 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ 𝑉 β‰  βˆ…)
8 elnnne0 12427 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‰) ∈ β„• ↔ ((β™―β€˜π‘‰) ∈ β„•0 ∧ (β™―β€˜π‘‰) β‰  0))
9 df-ne 2944 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘‰) β‰  1 ↔ Β¬ (β™―β€˜π‘‰) = 1)
10 eluz2b3 12847 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ↔ ((β™―β€˜π‘‰) ∈ β„• ∧ (β™―β€˜π‘‰) β‰  1))
11 hash2prde 14369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) = 2) β†’ βˆƒπ‘Žβˆƒπ‘(π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}))
12 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 π‘Ž ∈ V
1312a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ π‘Ž ∈ V)
14 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑏 ∈ V
1514a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ 𝑏 ∈ V)
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ π‘Ž β‰  𝑏)
1713, 15, 163jca 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘Ž β‰  𝑏 β†’ (π‘Ž ∈ V ∧ 𝑏 ∈ V ∧ π‘Ž β‰  𝑏))
18 frgrreggt1.v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑉 = (Vtxβ€˜πΊ)
1918eqeq1i 2741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑉 = {π‘Ž, 𝑏} ↔ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏})
2019biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑉 = {π‘Ž, 𝑏} β†’ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏})
21 nfrgr2v 29216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((π‘Ž ∈ V ∧ 𝑏 ∈ V ∧ π‘Ž β‰  𝑏) ∧ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏}) β†’ 𝐺 βˆ‰ FriendGraph )
2217, 20, 21syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ 𝐺 βˆ‰ FriendGraph )
23 df-nel 3050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝐺 βˆ‰ FriendGraph ↔ Β¬ 𝐺 ∈ FriendGraph )
2422, 23sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ Β¬ 𝐺 ∈ FriendGraph )
2524pm2.21d 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2625com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2726exlimivv 1935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (βˆƒπ‘Žβˆƒπ‘(π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2811, 27syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) = 2) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2928ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3029com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑉 ∈ Fin β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3130com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3231a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
33323imp 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
3433com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘‰) = 2 β†’ (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
35 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (VtxDegβ€˜πΊ) = (VtxDegβ€˜πΊ)
3618, 35rusgrprop0 28515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾))
37 eluz2gt1 12845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ 1 < (β™―β€˜π‘‰))
3837anim1ci 616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ) β†’ (𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)))
3918vdgn0frgrv2 29239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) β†’ (1 < (β™―β€˜π‘‰) β†’ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
4039impancom 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)) β†’ (𝑣 ∈ 𝑉 β†’ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
4140ralrimiv 3142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)) β†’ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
42 eqeq2 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (𝐾 = 0 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 0))
4342ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0))
44 r19.26 3114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
45 nne 2947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 (Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 0)
4645bicomi 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ↔ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
4746anbi1i 624 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
48 ancom 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
49 pm3.24 403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Β¬ (((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
5049bifal 1557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βŠ₯)
5147, 48, 503bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ((((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βŠ₯)
5251ralbii 3096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βˆ€π‘£ ∈ 𝑉 βŠ₯)
53 r19.3rzv 4456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (𝑉 β‰  βˆ… β†’ (βŠ₯ ↔ βˆ€π‘£ ∈ 𝑉 βŠ₯))
54 falim 1558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
5553, 54syl6bir 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 (𝑉 β‰  βˆ… β†’ (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5655adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5756com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5852, 57sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5944, 58sylbir 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
6059ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
6143, 60syl6bi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6261com4t 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6338, 41, 623syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6463ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6564com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6665adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6766com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝐺 ∈ FriendGraph β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6867com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
69683ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
7036, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
7170impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
7271impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
7318frrusgrord 29285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1)))
7473imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1))
75 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝐾 = 2 β†’ 𝐾 = 2)
76 oveq1 7364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝐾 = 2 β†’ (𝐾 βˆ’ 1) = (2 βˆ’ 1))
7775, 76oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝐾 = 2 β†’ (𝐾 Β· (𝐾 βˆ’ 1)) = (2 Β· (2 βˆ’ 1)))
7877oveq1d 7372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝐾 = 2 β†’ ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) = ((2 Β· (2 βˆ’ 1)) + 1))
79 2m1e1 12279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (2 βˆ’ 1) = 1
8079oveq2i 7368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (2 Β· (2 βˆ’ 1)) = (2 Β· 1)
81 2t1e2 12316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (2 Β· 1) = 2
8280, 81eqtri 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (2 Β· (2 βˆ’ 1)) = 2
8382oveq1i 7367 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((2 Β· (2 βˆ’ 1)) + 1) = (2 + 1)
84 2p1e3 12295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2 + 1) = 3
8583, 84eqtri 2764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((2 Β· (2 βˆ’ 1)) + 1) = 3
8678, 85eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝐾 = 2 β†’ ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) = 3)
8786eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝐾 = 2 β†’ ((β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) ↔ (β™―β€˜π‘‰) = 3))
88 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
8988ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9089com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘‰) = 3 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9187, 90syl6bi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝐾 = 2 β†’ ((β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
9274, 91syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 2 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
9318frgrreg 29338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐾 = 0 ∨ 𝐾 = 2)))
9493imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 0 ∨ 𝐾 = 2))
9572, 92, 94mpjaod 858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9695exp32 421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9796com34 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9897com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9998exp4c 433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
10099com34 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
101100com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
102101ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑉 ∈ Fin β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
103102com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
104103com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1051043imp 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
106105com3r 87 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Β¬ (β™―β€˜π‘‰) = 2 β†’ (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
10734, 106pm2.61i 182 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1081073exp 1119 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
10910, 108sylbir 234 . . . . . . . . . . . . . . . . . . . . . . 23 (((β™―β€˜π‘‰) ∈ β„• ∧ (β™―β€˜π‘‰) β‰  1) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
110109ex 413 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘‰) ∈ β„• β†’ ((β™―β€˜π‘‰) β‰  1 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1119, 110biimtrrid 242 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘‰) ∈ β„• β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
112111com25 99 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‰) ∈ β„• β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1138, 112sylbir 234 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‰) ∈ β„•0 ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
114113ex 413 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))))
115114impcomd 412 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
116115com14 96 . . . . . . . . . . . . . . . 16 (𝑉 β‰  βˆ… β†’ ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1177, 116mpcom 38 . . . . . . . . . . . . . . 15 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
118117ex 413 . . . . . . . . . . . . . 14 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
119118com14 96 . . . . . . . . . . . . 13 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1204, 119biimtrrid 242 . . . . . . . . . . . 12 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
121120com24 95 . . . . . . . . . . 11 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1221213imp 1111 . . . . . . . . . 10 (((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
123122com25 99 . . . . . . . . 9 (((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) β†’ (𝐺 RegUSGraph 𝐾 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
124123imp 407 . . . . . . . 8 ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
125124com14 96 . . . . . . 7 (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1261253imp 1111 . . . . . 6 ((Β¬ (β™―β€˜π‘‰) = 0 ∧ Β¬ (β™―β€˜π‘‰) = 1 ∧ Β¬ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
1273, 126sylbi 216 . . . . 5 (Β¬ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
1282, 127pm2.61i 182 . . . 4 ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
1291283exp1 1352 . . 3 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1301, 129mpcom 38 . 2 (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
1311303imp21 1114 1 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541  βŠ₯wfal 1553  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2943   βˆ‰ wnel 3049  βˆ€wral 3064  Vcvv 3445  βˆ…c0 4282  {cpr 4588   class class class wbr 5105  β€˜cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054   Β· cmul 11056   < clt 11189   βˆ’ cmin 11385  β„•cn 12153  2c2 12208  3c3 12209  β„•0cn0 12413  β„•0*cxnn0 12485  β„€β‰₯cuz 12763  β™―chash 14230  Vtxcvtx 27947  USGraphcusgr 28100  VtxDegcvtxdg 28413   RegUSGraph crusgr 28504   FriendGraph cfrgr 29202
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-ec 8650  df-qs 8654  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-reps 14657  df-csh 14677  df-s2 14737  df-s3 14738  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-dvds 16137  df-gcd 16375  df-prm 16548  df-phi 16638  df-vtx 27949  df-iedg 27950  df-edg 27999  df-uhgr 28009  df-ushgr 28010  df-upgr 28033  df-umgr 28034  df-uspgr 28101  df-usgr 28102  df-fusgr 28265  df-nbgr 28281  df-vtxdg 28414  df-rgr 28505  df-rusgr 28506  df-wlks 28547  df-wlkson 28548  df-trls 28640  df-trlson 28641  df-pths 28664  df-spths 28665  df-pthson 28666  df-spthson 28667  df-wwlks 28775  df-wwlksn 28776  df-wwlksnon 28777  df-wspthsn 28778  df-wspthsnon 28779  df-clwwlk 28926  df-clwwlkn 28969  df-clwwlknon 29032  df-conngr 29131  df-frgr 29203
This theorem is referenced by:  frgrregord13  29340
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