Axiom ax-dc 9870

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 9945. 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 1536	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1536	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1536	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5068	. . . . 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 5558	. . . 4 class ran 𝑥
11	6	cdm 5557	. . . 4 class dom 𝑥
12	10, 11	wss 3938	. . 3 wff ran 𝑥 ⊆ dom 𝑥
13	9, 12	wa 398	. 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14		vn	. . . . . . 7 setvar 𝑛
15	14	cv 1536	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1536	. . . . . 6 class 𝑓
18	15, 17	cfv 6357	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6195	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6357	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5068	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7582	. . . 4 class ω
23	21, 14, 22	wral 3140	. . 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  9871  axdc2lem  9872