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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 29582 | . 2 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺)) | |
2 | 1 | ssriv 3982 | 1 ⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺) |
Colors of variables: wff setvar class |
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 |
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