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Theorem vdgn1frgrav2 26319
Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
Assertion
Ref Expression
vdgn1frgrav2 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrav2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26285 . . . . . 6 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
21anim1i 589 . . . . 5 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (𝑉 USGrph 𝐸𝑁𝑉))
32adantr 479 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸𝑁𝑉))
4 vdusgraval 26200 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
53, 4syl 17 . . 3 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}))
6 3cyclfrgrarn2 26307 . . . . . 6 ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
76adantlr 746 . . . . 5 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
8 preq1 4211 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
98eleq1d 2671 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
10 preq2 4212 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1110eleq1d 2671 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
129, 113anbi13d 1392 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
1312anbi2d 735 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
14132rexbidv 3038 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))))
1514rspcva 3279 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
161adantl 480 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑉 USGrph 𝐸)
17 simplr 787 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑁𝑉)
18 simplll 793 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → 𝑏𝑐)
19 3simpb 1051 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
2019ad3antlr 762 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
21 usgra2edg1 25678 . . . . . . . . . . . . . . . . 17 (((𝑉 USGrph 𝐸𝑁𝑉𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
2216, 17, 18, 20, 21syl31anc 1320 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
2322a1d 25 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) ∧ 𝑉 FriendGrph 𝐸) → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
2423ex 448 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) ∧ 𝑁𝑉) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))
2524ex 448 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2625a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
2726rexlimivv 3017 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2815, 27syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
2928ex 448 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁𝑉 → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))))))
3029pm2.43a 51 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑉 FriendGrph 𝐸 → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3130com24 92 . . . . . . 7 (𝑁𝑉 → (1 < (#‘𝑉) → (𝑉 FriendGrph 𝐸 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3231com3r 84 . . . . . 6 (𝑉 FriendGrph 𝐸 → (𝑁𝑉 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))))
3332imp31 446 . . . . 5 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
347, 33mpd 15 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥))
35 usgrav 25633 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3635simprd 477 . . . . . . . . . 10 (𝑉 USGrph 𝐸𝐸 ∈ V)
37 dmexg 6966 . . . . . . . . . 10 (𝐸 ∈ V → dom 𝐸 ∈ V)
381, 36, 373syl 18 . . . . . . . . 9 (𝑉 FriendGrph 𝐸 → dom 𝐸 ∈ V)
3938adantr 479 . . . . . . . 8 ((𝑉 FriendGrph 𝐸𝑁𝑉) → dom 𝐸 ∈ V)
4039adantr 479 . . . . . . 7 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ V)
41 rabexg 4734 . . . . . . 7 (dom 𝐸 ∈ V → {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V)
42 hash1snb 13020 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
4340, 41, 423syl 18 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖}))
44 reusn 4205 . . . . . 6 (∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥) ↔ ∃𝑖{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} = {𝑖})
4543, 44syl6bbr 276 . . . . 5 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4645necon3abid 2817 . . . 4 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸𝑥)))
4734, 46mpbird 245 . . 3 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≠ 1)
485, 47eqnetrd 2848 . 2 (((𝑉 FriendGrph 𝐸𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1)
4948ex 448 1 ((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  wrex 2896  ∃!wreu 2897  {crab 2899  Vcvv 3172  {csn 4124  {cpr 4126   class class class wbr 4577  dom cdm 5028  ran crn 5029  cfv 5790  (class class class)co 6527  1c1 9793   < clt 9930  #chash 12934   USGrph cusg 25625   VDeg cvdg 26186   FriendGrph cfrgra 26281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-xadd 11779  df-fz 12153  df-hash 12935  df-usgra 25628  df-vdgr 26187  df-frgra 26282
This theorem is referenced by:  vdgfrgragt2  26320  vdgn1frgrav3  26321
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