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Axiom ax-dc 9603
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 9678. 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 1600 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1600 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1600 . . . . . 6 class 𝑥
72, 4, 6wbr 4886 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1823 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1823 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5356 . . . 4 class ran 𝑥
116cdm 5355 . . . 4 class dom 𝑥
1210, 11wss 3792 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 386 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1600 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1600 . . . . . 6 class 𝑓
1815, 17cfv 6135 . . . . 5 class (𝑓𝑛)
1915csuc 5978 . . . . . 6 class suc 𝑛
2019, 17cfv 6135 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 4886 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7343 . . . 4 class ω
2321, 14, 22wral 3090 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1823 . 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  9604  axdc2lem  9605
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