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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfacycgr1 | Structured version Visualization version GIF version |
Description: An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.) |
Ref | Expression |
---|---|
dfacycgr1 | β’ AcyclicGraph = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-acycgr 34432 | . 2 β’ AcyclicGraph = {π β£ Β¬ βπβπ(π(Cyclesβπ)π β§ π β β )} | |
2 | 2exanali 1861 | . . . 4 β’ (Β¬ βπβπ(π(Cyclesβπ)π β§ Β¬ π = β ) β βπβπ(π(Cyclesβπ)π β π = β )) | |
3 | df-ne 2939 | . . . . . 6 β’ (π β β β Β¬ π = β ) | |
4 | 3 | anbi2i 621 | . . . . 5 β’ ((π(Cyclesβπ)π β§ π β β ) β (π(Cyclesβπ)π β§ Β¬ π = β )) |
5 | 4 | 2exbii 1849 | . . . 4 β’ (βπβπ(π(Cyclesβπ)π β§ π β β ) β βπβπ(π(Cyclesβπ)π β§ Β¬ π = β )) |
6 | 2, 5 | xchnxbir 332 | . . 3 β’ (Β¬ βπβπ(π(Cyclesβπ)π β§ π β β ) β βπβπ(π(Cyclesβπ)π β π = β )) |
7 | 6 | abbii 2800 | . 2 β’ {π β£ Β¬ βπβπ(π(Cyclesβπ)π β§ π β β )} = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} |
8 | 1, 7 | eqtri 2758 | 1 β’ AcyclicGraph = {π β£ βπβπ(π(Cyclesβπ)π β π = β )} |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 βwal 1537 = wceq 1539 βwex 1779 {cab 2707 β wne 2938 β c0 4321 class class class wbr 5147 βcfv 6542 Cyclesccycls 29309 AcyclicGraphcacycgr 34431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-ne 2939 df-acycgr 34432 |
This theorem is referenced by: isacycgr1 34435 |
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