Theorem clwlkswks 29809

Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)

Assertion

Ref	Expression
clwlkswks	⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

Proof of Theorem clwlkswks

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkwlk 29808	. 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2	1	ssriv 3999	1 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

Syntax hints:   ⊆ wss 3963  ‘cfv 6563  Walkscwlks 29629  ClWalkscclwlks 29803

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-sep 5302  ax-nul 5312  ax-pr 5438

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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-iota 6516  df-fun 6565  df-fv 6571  df-clwlks 29804

This theorem is referenced by:  0clwlk0  30161  clwlknon2num  30397  numclwlk1lem2  30399