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

Step	Hyp	Ref	Expression
1		vy	. . . . . . 7
2	1	cv 1686	. . . . . 6
3		vz	. . . . . . 7
4	3	cv 1686	. . . . . 6
5		vx	. . . . . . 7
6	5	cv 1686	. . . . . 6
7	2, 4, 6	wbr 4267	. . . . 5
8	7, 3	wex 1581	. . . 4
9	8, 1	wex 1581	. . 3
10	6	crn 4812	. . . 4
11	6	cdm 4811	. . . 4
12	10, 11	wss 3305	. . 3
13	9, 12	wa 362	. 2
14		vn	. . . . . . 7
15	14	cv 1686	. . . . . 6
16		vf	. . . . . . 7
17	16	cv 1686	. . . . . 6
18	15, 17	cfv 5390	. . . . 5
19	15	csuc 4692	. . . . . 6
20	19, 17	cfv 5390	. . . . 5
21	18, 20, 6	wbr 4267	. . . 4
22		com 6446	. . . 4
23	21, 14, 22	wral 2694	. . 3
24	23, 16	wex 1581	. 2
25	13, 24	wi 4	1

This axiom is referenced by:  dcomex  8563  axdc2lem  8564