MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkswks Structured version   Visualization version   GIF version

Theorem clwlkswks 28432
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.
StepHypRef Expression
1 clwlkwlk 28431 . 2 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
21ssriv 3936 1 (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
Colors of variables: wff setvar class
Syntax hints:  wss 3898  cfv 6479  Walkscwlks 28252  ClWalkscclwlks 28426
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 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-clwlks 28427
This theorem is referenced by:  0clwlk0  28784  clwlknon2num  29020  numclwlk1lem2  29022
  Copyright terms: Public domain W3C validator