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Axiom ax-dc 8752
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 8827. 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 1369 . . . . . 6
3 vz . . . . . . 7
43cv 1369 . . . . . 6
5 vx . . . . . . 7
65cv 1369 . . . . . 6
72, 4, 6wbr 4409 . . . . 5
87, 3wex 1587 . . . 4
98, 1wex 1587 . . 3
106crn 4958 . . . 4
116cdm 4957 . . . 4
1210, 11wss 3442 . . 3
139, 12wa 369 . 2
14 vn . . . . . . 7
1514cv 1369 . . . . . 6
16 vf . . . . . . 7
1716cv 1369 . . . . . 6
1815, 17cfv 5537 . . . . 5
1915csuc 4838 . . . . . 6
2019, 17cfv 5537 . . . . 5
2118, 20, 6wbr 4409 . . . 4
22 com 6609 . . . 4
2321, 14, 22wral 2800 . . 3
2423, 16wex 1587 . 2
2513, 24wi 4 1
Colors of variables: wff setvar class
This axiom is referenced by:  dcomex  8753  axdc2lem  8754
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