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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 6839 | . . . . 5 β’ (π = πΊ β (Cyclesβπ) = (CyclesβπΊ)) | |
2 | 1 | breqd 5114 | . . . 4 β’ (π = πΊ β (π(Cyclesβπ)π β π(CyclesβπΊ)π)) |
3 | 2 | imbi1d 341 | . . 3 β’ (π = πΊ β ((π(Cyclesβπ)π β π = β ) β (π(CyclesβπΊ)π β π = β ))) |
4 | 3 | 2albidv 1926 | . 2 β’ (π = πΊ β (βπβπ(π(Cyclesβπ)π β π = β ) β βπβπ(π(CyclesβπΊ)π β π = β ))) |
5 | dfacycgr1 33550 | . 2 β’ AcyclicGraph = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} | |
6 | 4, 5 | elab2g 3630 | 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 1539 = wceq 1541 β wcel 2106 β c0 4280 class class class wbr 5103 βcfv 6493 Cyclesccycls 28562 AcyclicGraphcacycgr 33548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-acycgr 33549 |
This theorem is referenced by: acycgrcycl 33553 acycgr1v 33555 |
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