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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.) |
Ref | Expression |
---|---|
ax-dc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vy | . . . . . . 7 | |
2 | 1 | cv 1394 | . . . . . 6 |
3 | vz | . . . . . . 7 | |
4 | 3 | cv 1394 | . . . . . 6 |
5 | vx | . . . . . . 7 | |
6 | 5 | cv 1394 | . . . . . 6 |
7 | 2, 4, 6 | wbr 4452 | . . . . 5 |
8 | 7, 3 | wex 1612 | . . . 4 |
9 | 8, 1 | wex 1612 | . . 3 |
10 | 6 | crn 5005 | . . . 4 |
11 | 6 | cdm 5004 | . . . 4 |
12 | 10, 11 | wss 3475 | . . 3 |
13 | 9, 12 | wa 369 | . 2 |
14 | vn | . . . . . . 7 | |
15 | 14 | cv 1394 | . . . . . 6 |
16 | vf | . . . . . . 7 | |
17 | 16 | cv 1394 | . . . . . 6 |
18 | 15, 17 | cfv 5593 | . . . . 5 |
19 | 15 | csuc 4885 | . . . . . 6 |
20 | 19, 17 | cfv 5593 | . . . . 5 |
21 | 18, 20, 6 | wbr 4452 | . . . 4 |
22 | com 6700 | . . . 4 | |
23 | 21, 14, 22 | wral 2807 | . . 3 |
24 | 23, 16 | wex 1612 | . 2 |
25 | 13, 24 | wi 4 | 1 |
Colors of variables: wff setvar class |
This axiom is referenced by: dcomex 8848 axdc2lem 8849 |
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