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Axiom ax-dc 9605
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 9680. 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 4888 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1823 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1823 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5358 . . . 4 class ran 𝑥
116cdm 5357 . . . 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 6137 . . . . 5 class (𝑓𝑛)
1915csuc 5980 . . . . . 6 class suc 𝑛
2019, 17cfv 6137 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 4888 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7345 . . . 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  9606  axdc2lem  9607
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