Axiom ax-dc 10334

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 10409. 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 1540	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1540	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1540	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5091	. . . . 5 wff 𝑦𝑥𝑧
8	7, 3	wex 1780	. . . 4 wff ∃𝑧 𝑦𝑥𝑧
9	8, 1	wex 1780	. . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧
10	6	crn 5617	. . . 4 class ran 𝑥
11	6	cdm 5616	. . . 4 class dom 𝑥
12	10, 11	wss 3902	. . 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 1540	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1540	. . . . . 6 class 𝑓
18	15, 17	cfv 6481	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6308	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6481	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5091	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7796	. . . 4 class ω
23	21, 14, 22	wral 3047	. . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
24	23, 16	wex 1780	. 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  10335  axdc2lem  10336