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Theorem wlkon2n0 29356
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 2731 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
21wlkonprop 29348 . . . 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 6900 . . . . . . . . 9 ((♯‘𝐹) = 0 → ((𝑃‘(♯‘𝐹)) = 𝐵 ↔ (𝑃‘0) = 𝐵))
43anbi2d 628 . . . . . . . 8 ((♯‘𝐹) = 0 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵) ↔ ((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵)))
5 eqtr2 2755 . . . . . . . . 9 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → 𝐴 = 𝐵)
6 nne 2943 . . . . . . . . 9 𝐴𝐵𝐴 = 𝐵)
75, 6sylibr 233 . . . . . . . 8 (((𝑃‘0) = 𝐴 ∧ (𝑃‘0) = 𝐵) → ¬ 𝐴𝐵)
84, 7syl6bi 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 2954 . 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 1540  wcel 2105  wne 2939  Vcvv 3473   class class class wbr 5148  cfv 6543  (class class class)co 7412  0cc0 11116  chash 14297  Vtxcvtx 28689  Walkscwlks 29286  WalksOncwlkson 29287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-wlkson 29290
This theorem is referenced by:  conngrv2edg  29881
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