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Theorem wlkon2n0 29684
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 2737 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
21wlkonprop 29676 . . . 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 6915 . . . . . . . . 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 2761 . . . . . . . . 9 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
6 nne 2944 . . . . . . . . 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 1131 . . . . 5 ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) → ((♯‘𝐹) = 0 → ¬ 𝐴𝐵))
11103ad2ant3 1136 . . . 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 2955 . 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 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480   class class class wbr 5143  cfv 6561  (class class class)co 7431  0cc0 11155  chash 14369  Vtxcvtx 29013  Walkscwlks 29614  WalksOncwlkson 29615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-wlkson 29618
This theorem is referenced by:  conngrv2edg  30214
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