MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-dc Structured version   Visualization version   GIF version

Axiom ax-dc 9866
Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 9941. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
ax-dc ((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
Distinct variable group:   𝑓,𝑛,𝑥,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-dc
StepHypRef Expression
1 vy . . . . . . 7 setvar 𝑦
21cv 1537 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1537 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1537 . . . . . 6 class 𝑥
72, 4, 6wbr 5052 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1781 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1781 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5543 . . . 4 class ran 𝑥
116cdm 5542 . . . 4 class dom 𝑥
1210, 11wss 3919 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 399 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1537 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1537 . . . . . 6 class 𝑓
1815, 17cfv 6343 . . . . 5 class (𝑓𝑛)
1915csuc 6180 . . . . . 6 class suc 𝑛
2019, 17cfv 6343 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 5052 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7574 . . . 4 class ω
2321, 14, 22wral 3133 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1781 . 2 wff 𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2513, 24wi 4 1 wff ((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
Colors of variables: wff setvar class
This axiom is referenced by:  dcomex  9867  axdc2lem  9868
  Copyright terms: Public domain W3C validator