Axiom ax-dc 10486

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 10561. 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 1539	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1539	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1539	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5143	. . . . 5 wff 𝑦𝑥𝑧
8	7, 3	wex 1779	. . . 4 wff ∃𝑧 𝑦𝑥𝑧
9	8, 1	wex 1779	. . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧
10	6	crn 5686	. . . 4 class ran 𝑥
11	6	cdm 5685	. . . 4 class dom 𝑥
12	10, 11	wss 3951	. . 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 1539	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1539	. . . . . 6 class 𝑓
18	15, 17	cfv 6561	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6386	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6561	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5143	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7887	. . . 4 class ω
23	21, 14, 22	wral 3061	. . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
24	23, 16	wex 1779	. 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  10487  axdc2lem  10488