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Axiom ax-dc 8562
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 8637. 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
Distinct variable group:   , , , ,

Detailed syntax breakdown of Axiom ax-dc
StepHypRef Expression
1 vy . . . . . . 7
21cv 1686 . . . . . 6
3 vz . . . . . . 7
43cv 1686 . . . . . 6
5 vx . . . . . . 7
65cv 1686 . . . . . 6
72, 4, 6wbr 4267 . . . . 5
87, 3wex 1581 . . . 4
98, 1wex 1581 . . 3
106crn 4812 . . . 4
116cdm 4811 . . . 4
1210, 11wss 3305 . . 3
139, 12wa 362 . 2
14 vn . . . . . . 7
1514cv 1686 . . . . . 6
16 vf . . . . . . 7
1716cv 1686 . . . . . 6
1815, 17cfv 5390 . . . . 5
1915csuc 4692 . . . . . 6
2019, 17cfv 5390 . . . . 5
2118, 20, 6wbr 4267 . . . 4
22 com 6446 . . . 4
2321, 14, 22wral 2694 . . 3
2423, 16wex 1581 . 2
2513, 24wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  dcomex  8563  axdc2lem  8564
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