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Mirrors > Home > MPE Home > Th. List > clwlkswks | Structured version Visualization version GIF version |
Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
clwlkswks | ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkwlk 28431 | . 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺)) | |
2 | 1 | ssriv 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 |
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