Axiom ax-dc 8605

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 8680. 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	|- ( ( E. y E. z y x z /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) )

Distinct variable group:   f, n, x, y, z

Detailed syntax breakdown of Axiom ax-dc

Step	Hyp	Ref	Expression
1		vy	. . . . . . 7 setvar y
2	1	cv 1363	. . . . . 6 class y
3		vz	. . . . . . 7 setvar z
4	3	cv 1363	. . . . . 6 class z
5		vx	. . . . . . 7 setvar x
6	5	cv 1363	. . . . . 6 class x
7	2, 4, 6	wbr 4282	. . . . 5 wff y x z
8	7, 3	wex 1591	. . . 4 wff E. z y x z
9	8, 1	wex 1591	. . 3 wff E. y E. z y x z
10	6	crn 4830	. . . 4 class ran x
11	6	cdm 4829	. . . 4 class dom x
12	10, 11	wss 3318	. . 3 wff ran x C_ dom x
13	9, 12	wa 369	. 2 wff ( E. y E. z y x z /\ ran x C_ dom x )
14		vn	. . . . . . 7 setvar n
15	14	cv 1363	. . . . . 6 class n
16		vf	. . . . . . 7 setvar f
17	16	cv 1363	. . . . . 6 class f
18	15, 17	cfv 5408	. . . . 5 class ( f ` n )
19	15	csuc 4710	. . . . . 6 class suc n
20	19, 17	cfv 5408	. . . . 5 class ( f ` suc n )
21	18, 20, 6	wbr 4282	. . . 4 wff ( f ` n ) x ( f ` suc n )
22		com 6467	. . . 4 class om
23	21, 14, 22	wral 2707	. . 3 wff A. n e. om ( f ` n ) x ( f ` suc n )
24	23, 16	wex 1591	. 2 wff E. f A. n e. om ( f ` n ) x ( f ` suc n )
25	13, 24	wi 4	1 wff ( ( E. y E. z y x z /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) )

This axiom is referenced by:  dcomex  8606  axdc2lem  8607