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Axiom ax-dc 10486
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 10561. 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 1539 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1539 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1539 . . . . . 6 class 𝑥
72, 4, 6wbr 5143 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1779 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1779 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5686 . . . 4 class ran 𝑥
116cdm 5685 . . . 4 class dom 𝑥
1210, 11wss 3951 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 395 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1539 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1539 . . . . . 6 class 𝑓
1815, 17cfv 6561 . . . . 5 class (𝑓𝑛)
1915csuc 6386 . . . . . 6 class suc 𝑛
2019, 17cfv 6561 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 5143 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7887 . . . 4 class ω
2321, 14, 22wral 3061 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1779 . 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  10487  axdc2lem  10488
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