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 10483
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 10558. 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 1535 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1535 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1535 . . . . . 6 class 𝑥
72, 4, 6wbr 5147 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1775 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1775 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5689 . . . 4 class ran 𝑥
116cdm 5688 . . . 4 class dom 𝑥
1210, 11wss 3962 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 395 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1535 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1535 . . . . . 6 class 𝑓
1815, 17cfv 6562 . . . . 5 class (𝑓𝑛)
1915csuc 6387 . . . . . 6 class suc 𝑛
2019, 17cfv 6562 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 5147 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7886 . . . 4 class ω
2321, 14, 22wral 3058 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1775 . 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  10484  axdc2lem  10485
  Copyright terms: Public domain W3C validator