Axiom ax-dc 9151

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 9226. 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	⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (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 1474	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1474	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1474	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 4583	. . . . 5 wff 𝑦𝑥𝑧
8	7, 3	wex 1695	. . . 4 wff ∃𝑧 𝑦𝑥𝑧
9	8, 1	wex 1695	. . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧
10	6	crn 5039	. . . 4 class ran 𝑥
11	6	cdm 5038	. . . 4 class dom 𝑥
12	10, 11	wss 3540	. . 3 wff ran 𝑥 ⊆ dom 𝑥
13	9, 12	wa 383	. 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14		vn	. . . . . . 7 setvar 𝑛
15	14	cv 1474	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1474	. . . . . 6 class 𝑓
18	15, 17	cfv 5804	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 5642	. . . . . 6 class suc 𝑛
20	19, 17	cfv 5804	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 4583	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 6957	. . . 4 class ω
23	21, 14, 22	wral 2896	. . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
24	23, 16	wex 1695	. 2 wff ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
25	13, 24	wi 4	1 wff ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))

This axiom is referenced by:  dcomex  9152  axdc2lem  9153