Axiom ax-dc 8730

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 8805. 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 1369	. . . . . 6 class y
3		vz	. . . . . . 7 setvar z
4	3	cv 1369	. . . . . 6 class z
5		vx	. . . . . . 7 setvar x
6	5	cv 1369	. . . . . 6 class x
7	2, 4, 6	wbr 4403	. . . . 5 wff y x z
8	7, 3	wex 1587	. . . 4 wff E. z y x z
9	8, 1	wex 1587	. . 3 wff E. y E. z y x z
10	6	crn 4952	. . . 4 class ran x
11	6	cdm 4951	. . . 4 class dom x
12	10, 11	wss 3439	. . 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 1369	. . . . . 6 class n
16		vf	. . . . . . 7 setvar f
17	16	cv 1369	. . . . . 6 class f
18	15, 17	cfv 5529	. . . . 5 class ( f ` n )
19	15	csuc 4832	. . . . . 6 class suc n
20	19, 17	cfv 5529	. . . . 5 class ( f ` suc n )
21	18, 20, 6	wbr 4403	. . . 4 wff ( f ` n ) x ( f ` suc n )
22		com 6589	. . . 4 class om
23	21, 14, 22	wral 2799	. . 3 wff A. n e. om ( f ` n ) x ( f ` suc n )
24	23, 16	wex 1587	. 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  8731  axdc2lem  8732