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

Step	Hyp	Ref	Expression
1		vy	. . . . . . 7
2	1	cv 1369	. . . . . 6
3		vz	. . . . . . 7
4	3	cv 1369	. . . . . 6
5		vx	. . . . . . 7
6	5	cv 1369	. . . . . 6
7	2, 4, 6	wbr 4409	. . . . 5
8	7, 3	wex 1587	. . . 4
9	8, 1	wex 1587	. . 3
10	6	crn 4958	. . . 4
11	6	cdm 4957	. . . 4
12	10, 11	wss 3442	. . 3
13	9, 12	wa 369	. 2
14		vn	. . . . . . 7
15	14	cv 1369	. . . . . 6
16		vf	. . . . . . 7
17	16	cv 1369	. . . . . 6
18	15, 17	cfv 5537	. . . . 5
19	15	csuc 4838	. . . . . 6
20	19, 17	cfv 5537	. . . . 5
21	18, 20, 6	wbr 4409	. . . 4
22		com 6609	. . . 4
23	21, 14, 22	wral 2800	. . 3
24	23, 16	wex 1587	. 2
25	13, 24	wi 4	1

This axiom is referenced by:  dcomex  8753  axdc2lem  8754