Theorem clwlkswks 29706

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 29705	. 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2	1	ssriv 3950	1 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

Syntax hints:   ⊆ wss 3914  ‘cfv 6511  Walkscwlks 29524  ClWalkscclwlks 29700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-clwlks 29701

This theorem is referenced by:  0clwlk0  30061  clwlknon2num  30297  numclwlk1lem2  30299