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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 | ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-acycgr 32409 | . 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)} | |
2 | 2exanali 1859 | . . . 4 ⊢ (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)) | |
3 | df-ne 3016 | . . . . . 6 ⊢ (𝑓 ≠ ∅ ↔ ¬ 𝑓 = ∅) | |
4 | 3 | anbi2i 624 | . . . . 5 ⊢ ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅)) |
5 | 4 | 2exbii 1848 | . . . 4 ⊢ (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅)) |
6 | 2, 5 | xchnxbir 335 | . . 3 ⊢ (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)) |
7 | 6 | abbii 2885 | . 2 ⊢ {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)} = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)} |
8 | 1, 7 | eqtri 2843 | 1 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∃wex 1779 {cab 2798 ≠ wne 3015 ∅c0 4284 class class class wbr 5059 ‘cfv 6348 Cyclesccycls 27562 AcyclicGraphcacycgr 32408 |
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 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-ne 3016 df-acycgr 32409 |
This theorem is referenced by: isacycgr1 32412 |
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