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Theorem wspthsswwlkn 27718
 Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.)
Assertion
Ref Expression
wspthsswwlkn (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)

Proof of Theorem wspthsswwlkn
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wspthnp 27650 . . 3 (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤))
21simp2d 1140 . 2 (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺))
32ssriv 3919 1 (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881   class class class wbr 5031  ‘cfv 6327  (class class class)co 7140  ℕ0cn0 11892  SPathscspths 27516   WWalksN cwwlksn 27626   WSPathsN cwwspthsn 27628 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-wwlksn 27631  df-wspthsn 27633 This theorem is referenced by:  wspthnfi  27719  fusgreg2wsp  28135
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