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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 | ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6736 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Cycles‘𝑔) = (Cycles‘𝐺)) | |
2 | 1 | breqd 5079 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓(Cycles‘𝑔)𝑝 ↔ 𝑓(Cycles‘𝐺)𝑝)) |
3 | 2 | anbi1d 633 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
4 | 3 | 2exbidv 1932 | . . 3 ⊢ (𝑔 = 𝐺 → (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
5 | 4 | notbid 321 | . 2 ⊢ (𝑔 = 𝐺 → (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
6 | df-acycgr 32841 | . 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)} | |
7 | 5, 6 | elab2g 3602 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 ≠ wne 2941 ∅c0 4252 class class class wbr 5068 ‘cfv 6398 Cyclesccycls 27896 AcyclicGraphcacycgr 32840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-iota 6356 df-fv 6406 df-acycgr 32841 |
This theorem is referenced by: acycgr0v 32846 acycgr2v 32848 acycgrislfgr 32850 umgracycusgr 32852 cusgracyclt3v 32854 acycgrsubgr 32856 |
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