Axiom ax-dc 10406

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 10481. 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 5110	. . . . 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 5642	. . . 4 class ran 𝑥
11	6	cdm 5641	. . . 4 class dom 𝑥
12	10, 11	wss 3917	. . 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 6514	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6337	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6514	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5110	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7845	. . . 4 class ω
23	21, 14, 22	wral 3045	. . 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  10407  axdc2lem  10408