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Axiom ax-dc 9228
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 9303. 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 1479 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1479 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1479 . . . . . 6 class 𝑥
72, 4, 6wbr 4623 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1701 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1701 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5085 . . . 4 class ran 𝑥
116cdm 5084 . . . 4 class dom 𝑥
1210, 11wss 3560 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 384 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1479 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1479 . . . . . 6 class 𝑓
1815, 17cfv 5857 . . . . 5 class (𝑓𝑛)
1915csuc 5694 . . . . . 6 class suc 𝑛
2019, 17cfv 5857 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 4623 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7027 . . . 4 class ω
2321, 14, 22wral 2908 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1701 . 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  9229  axdc2lem  9230
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