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 9025
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 9100. 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 1473 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1473 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1473 . . . . . 6 class 𝑥
72, 4, 6wbr 4481 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1694 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1694 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 4933 . . . 4 class ran 𝑥
116cdm 4932 . . . 4 class dom 𝑥
1210, 11wss 3444 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 382 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1473 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1473 . . . . . 6 class 𝑓
1815, 17cfv 5689 . . . . 5 class (𝑓𝑛)
1915csuc 5532 . . . . . 6 class suc 𝑛
2019, 17cfv 5689 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 4481 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 6831 . . . 4 class ω
2321, 14, 22wral 2800 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1694 . 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  9026  axdc2lem  9027
  Copyright terms: Public domain W3C validator