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Axiom ax-dc 8847
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 8922. 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 1394 . . . . . 6
3 vz . . . . . . 7
43cv 1394 . . . . . 6
5 vx . . . . . . 7
65cv 1394 . . . . . 6
72, 4, 6wbr 4452 . . . . 5
87, 3wex 1612 . . . 4
98, 1wex 1612 . . 3
106crn 5005 . . . 4
116cdm 5004 . . . 4
1210, 11wss 3475 . . 3
139, 12wa 369 . 2
14 vn . . . . . . 7
1514cv 1394 . . . . . 6
16 vf . . . . . . 7
1716cv 1394 . . . . . 6
1815, 17cfv 5593 . . . . 5
1915csuc 4885 . . . . . 6
2019, 17cfv 5593 . . . . 5
2118, 20, 6wbr 4452 . . . 4
22 com 6700 . . . 4
2321, 14, 22wral 2807 . . 3
2423, 16wex 1612 . 2
2513, 24wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  dcomex  8848  axdc2lem  8849
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