Axiom ax-dc 9857

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 9932. 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 1537	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1537	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1537	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5030	. . . . 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 5520	. . . 4 class ran 𝑥
11	6	cdm 5519	. . . 4 class dom 𝑥
12	10, 11	wss 3881	. . 3 wff ran 𝑥 ⊆ dom 𝑥
13	9, 12	wa 399	. 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14		vn	. . . . . . 7 setvar 𝑛
15	14	cv 1537	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1537	. . . . . 6 class 𝑓
18	15, 17	cfv 6324	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6161	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6324	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5030	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7560	. . . 4 class ω
23	21, 14, 22	wral 3106	. . 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  9858  axdc2lem  9859