Theorem clwlkswks 29583

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 29582	. 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2	1	ssriv 3982	1 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

Syntax hints:   ⊆ wss 3944  ‘cfv 6542  Walkscwlks 29403  ClWalkscclwlks 29577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-clwlks 29578

This theorem is referenced by:  0clwlk0  29935  clwlknon2num  30171  numclwlk1lem2  30173