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Axiom ax-dc 9306
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 9381. 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 1522 . . . . . 6 class 𝑦
3 vz . . . . . . 7 setvar 𝑧
43cv 1522 . . . . . 6 class 𝑧
5 vx . . . . . . 7 setvar 𝑥
65cv 1522 . . . . . 6 class 𝑥
72, 4, 6wbr 4685 . . . . 5 wff 𝑦𝑥𝑧
87, 3wex 1744 . . . 4 wff 𝑧 𝑦𝑥𝑧
98, 1wex 1744 . . 3 wff 𝑦𝑧 𝑦𝑥𝑧
106crn 5144 . . . 4 class ran 𝑥
116cdm 5143 . . . 4 class dom 𝑥
1210, 11wss 3607 . . 3 wff ran 𝑥 ⊆ dom 𝑥
139, 12wa 383 . 2 wff (∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14 vn . . . . . . 7 setvar 𝑛
1514cv 1522 . . . . . 6 class 𝑛
16 vf . . . . . . 7 setvar 𝑓
1716cv 1522 . . . . . 6 class 𝑓
1815, 17cfv 5926 . . . . 5 class (𝑓𝑛)
1915csuc 5763 . . . . . 6 class suc 𝑛
2019, 17cfv 5926 . . . . 5 class (𝑓‘suc 𝑛)
2118, 20, 6wbr 4685 . . . 4 wff (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
22 com 7107 . . . 4 class ω
2321, 14, 22wral 2941 . . 3 wff 𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)
2423, 16wex 1744 . 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  9307  axdc2lem  9308
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