Axiom ax-dc 10357

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 10432. 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 1541	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1541	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1541	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5086	. . . . 5 wff 𝑦𝑥𝑧
8	7, 3	wex 1781	. . . 4 wff ∃𝑧 𝑦𝑥𝑧
9	8, 1	wex 1781	. . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧
10	6	crn 5623	. . . 4 class ran 𝑥
11	6	cdm 5622	. . . 4 class dom 𝑥
12	10, 11	wss 3890	. . 3 wff ran 𝑥 ⊆ dom 𝑥
13	9, 12	wa 395	. 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14		vn	. . . . . . 7 setvar 𝑛
15	14	cv 1541	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1541	. . . . . 6 class 𝑓
18	15, 17	cfv 6490	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6317	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6490	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5086	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7808	. . . 4 class ω
23	21, 14, 22	wral 3052	. . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
24	23, 16	wex 1781	. 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  10358  axdc2lem  10359