Axiom ax-dc 8836

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 8911. 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 1378	. . . . . 6 class y
3		vz	. . . . . . 7 setvar z
4	3	cv 1378	. . . . . 6 class z
5		vx	. . . . . . 7 setvar x
6	5	cv 1378	. . . . . 6 class x
7	2, 4, 6	wbr 4452	. . . . 5 wff y x z
8	7, 3	wex 1596	. . . 4 wff E. z y x z
9	8, 1	wex 1596	. . 3 wff E. y E. z y x z
10	6	crn 5005	. . . 4 class ran x
11	6	cdm 5004	. . . 4 class dom x
12	10, 11	wss 3481	. . 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 1378	. . . . . 6 class n
16		vf	. . . . . . 7 setvar f
17	16	cv 1378	. . . . . 6 class f
18	15, 17	cfv 5593	. . . . 5 class ( f ` n )
19	15	csuc 4885	. . . . . 6 class suc n
20	19, 17	cfv 5593	. . . . 5 class ( f ` suc n )
21	18, 20, 6	wbr 4452	. . . 4 wff ( f ` n ) x ( f ` suc n )
22		com 6694	. . . 4 class om
23	21, 14, 22	wral 2817	. . 3 wff A. n e. om ( f ` n ) x ( f ` suc n )
24	23, 16	wex 1596	. 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  8837  axdc2lem  8838