Axiom ax-dc 8829

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 8904. 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 1382	. . . . . 6 class y
3		vz	. . . . . . 7 setvar z
4	3	cv 1382	. . . . . 6 class z
5		vx	. . . . . . 7 setvar x
6	5	cv 1382	. . . . . 6 class x
7	2, 4, 6	wbr 4437	. . . . 5 wff y x z
8	7, 3	wex 1599	. . . 4 wff E. z y x z
9	8, 1	wex 1599	. . 3 wff E. y E. z y x z
10	6	crn 4990	. . . 4 class ran x
11	6	cdm 4989	. . . 4 class dom x
12	10, 11	wss 3461	. . 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 1382	. . . . . 6 class n
16		vf	. . . . . . 7 setvar f
17	16	cv 1382	. . . . . 6 class f
18	15, 17	cfv 5578	. . . . 5 class ( f ` n )
19	15	csuc 4870	. . . . . 6 class suc n
20	19, 17	cfv 5578	. . . . 5 class ( f ` suc n )
21	18, 20, 6	wbr 4437	. . . 4 wff ( f ` n ) x ( f ` suc n )
22		com 6685	. . . 4 class om
23	21, 14, 22	wral 2793	. . 3 wff A. n e. om ( f ` n ) x ( f ` suc n )
24	23, 16	wex 1599	. 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  8830  axdc2lem  8831