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Theorem wlkon2n0 29699
Description: The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.)
Assertion
Ref Expression
wlkon2n0 ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐴𝐵) → (♯‘𝐹) ≠ 0)

Proof of Theorem wlkon2n0
StepHypRef Expression
1 eqid 2735 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
21wlkonprop 29691 . . . 4 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
3 fveqeq2 6916 . . . . . . . . 9 ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
43anbi2d 630 . . . . . . . 8 ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
5 eqtr2 2759 . . . . . . . . 9 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
6 nne 2942 . . . . . . . . 9 𝐴𝐵𝐴 = 𝐵)
75, 6sylibr 234 . . . . . . . 8 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → ¬ 𝐴𝐵)
84, 7biimtrdi 253 . . . . . . 7 ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ¬ 𝐴𝐵))
98com12 32 . . . . . 6 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → ¬ 𝐴𝐵))
1093adant1 1129 . . . . 5 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → ¬ 𝐴𝐵))
11103ad2ant3 1134 . . . 4 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) → ((♯‘𝐹) = 0 → ¬ 𝐴𝐵))
122, 11syl 17 . . 3 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((♯‘𝐹) = 0 → ¬ 𝐴𝐵))
1312necon2ad 2953 . 2 (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → (𝐴𝐵 → (♯‘𝐹) ≠ 0))
1413imp 406 1 ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐴𝐵) → (♯‘𝐹) ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478   class class class wbr 5148  cfv 6563  (class class class)co 7431  0cc0 11153  chash 14366  Vtxcvtx 29028  Walkscwlks 29629  WalksOncwlkson 29630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-wlkson 29633
This theorem is referenced by:  conngrv2edg  30224
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