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Mirrors > Home > MPE Home > Th. List > Mathboxes > isacycgr1 | Structured version Visualization version GIF version |
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
isacycgr1 | β’ (πΊ β π β (πΊ β AcyclicGraph β βπβπ(π(CyclesβπΊ)π β π = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6902 | . . . . 5 β’ (π = πΊ β (Cyclesβπ) = (CyclesβπΊ)) | |
2 | 1 | breqd 5163 | . . . 4 β’ (π = πΊ β (π(Cyclesβπ)π β π(CyclesβπΊ)π)) |
3 | 2 | imbi1d 340 | . . 3 β’ (π = πΊ β ((π(Cyclesβπ)π β π = β ) β (π(CyclesβπΊ)π β π = β ))) |
4 | 3 | 2albidv 1918 | . 2 β’ (π = πΊ β (βπβπ(π(Cyclesβπ)π β π = β ) β βπβπ(π(CyclesβπΊ)π β π = β ))) |
5 | dfacycgr1 34795 | . 2 β’ AcyclicGraph = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} | |
6 | 4, 5 | elab2g 3671 | 1 β’ (πΊ β π β (πΊ β AcyclicGraph β βπβπ(π(CyclesβπΊ)π β π = β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 βwal 1531 = wceq 1533 β wcel 2098 β c0 4326 class class class wbr 5152 βcfv 6553 Cyclesccycls 29627 AcyclicGraphcacycgr 34793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-acycgr 34794 |
This theorem is referenced by: acycgrcycl 34798 acycgr1v 34800 |
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