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Mirrors > Home > MPE Home > Th. List > Mathboxes > isacycgr | Structured version Visualization version GIF version |
Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
isacycgr | β’ (πΊ β π β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6892 | . . . . . 6 β’ (π = πΊ β (Cyclesβπ) = (CyclesβπΊ)) | |
2 | 1 | breqd 5160 | . . . . 5 β’ (π = πΊ β (π(Cyclesβπ)π β π(CyclesβπΊ)π)) |
3 | 2 | anbi1d 631 | . . . 4 β’ (π = πΊ β ((π(Cyclesβπ)π β§ π β β ) β (π(CyclesβπΊ)π β§ π β β ))) |
4 | 3 | 2exbidv 1928 | . . 3 β’ (π = πΊ β (βπβπ(π(Cyclesβπ)π β§ π β β ) β βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
5 | 4 | notbid 318 | . 2 β’ (π = πΊ β (Β¬ βπβπ(π(Cyclesβπ)π β§ π β β ) β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
6 | df-acycgr 34134 | . 2 β’ AcyclicGraph = {π β£ Β¬ βπβπ(π(Cyclesβπ)π β§ π β β )} | |
7 | 5, 6 | elab2g 3671 | 1 β’ (πΊ β π β (πΊ β AcyclicGraph β Β¬ βπβπ(π(CyclesβπΊ)π β§ π β β ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 β wne 2941 β c0 4323 class class class wbr 5149 βcfv 6544 Cyclesccycls 29042 AcyclicGraphcacycgr 34133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-acycgr 34134 |
This theorem is referenced by: acycgr0v 34139 acycgr2v 34141 acycgrislfgr 34143 umgracycusgr 34145 cusgracyclt3v 34147 acycgrsubgr 34149 |
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