Theorem clwlkswks 30368

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 30367	. 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2	1	ssriv 3912	1 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

This proof depends on syntax axioms:   ⊆ wss 3895  ‘cfv 6538  Walkscwlks 30182  ClWalkscclwlks 30362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-clwlks 30363

This theorem is used by:  0clwlk0  30728  clwlknon2num  30974  numclwlk1lem2  30976