Theorem dfacycgr1 30783

Description: An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)

Assertion

Ref	Expression
dfacycgr1	⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}

Distinct variable group:   𝑓,𝑔,𝑝

Proof of Theorem dfacycgr1

Step	Hyp	Ref	Expression
1		df-acycgr 30782	. 2 ⊢ AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
2		2exanali 1893	. . . 4 ⊢ (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅))
3		df-ne 2960	. . . . . 6 ⊢ (𝑓 ≠ ∅ ↔ ¬ 𝑓 = ∅)
4	3	anbi2i 635	. . . . 5 ⊢ ((𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ (𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅))
5	4	2exbii 1882	. . . 4 ⊢ (∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ ¬ 𝑓 = ∅))
6	2, 5	xchnxbir 336	. . 3 ⊢ (¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅) ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅))
7	6	abbii 2830	. 2 ⊢ {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)} = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
8	1, 7	eqtri 2786	1 ⊢ AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  {cab 2741   ≠ wne 2959  ∅c0 4281   class class class wbr 5106  ‘cfv 6561  Cyclesccycls 30405  AcyclicGraphcacycgr 30781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ne 2960  df-acycgr 30782

This theorem is used by:  isacycgr1  30785