Axiom ax-dc 10448

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 10523. 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 1569	. . . . . 6 class 𝑦
3		vz	. . . . . . 7 setvar 𝑧
4	3	cv 1569	. . . . . 6 class 𝑧
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1569	. . . . . 6 class 𝑥
7	2, 4, 6	wbr 5114	. . . . 5 wff 𝑦𝑥𝑧
8	7, 3	wex 1812	. . . 4 wff ∃𝑧 𝑦𝑥𝑧
9	8, 1	wex 1812	. . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧
10	6	crn 5667	. . . 4 class ran 𝑥
11	6	cdm 5666	. . . 4 class dom 𝑥
12	10, 11	wss 3908	. . 3 wff ran 𝑥 ⊆ dom 𝑥
13	9, 12	wa 401	. 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)
14		vn	. . . . . . 7 setvar 𝑛
15	14	cv 1569	. . . . . 6 class 𝑛
16		vf	. . . . . . 7 setvar 𝑓
17	16	cv 1569	. . . . . 6 class 𝑓
18	15, 17	cfv 6543	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6369	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6543	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5114	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7871	. . . 4 class ω
23	21, 14, 22	wral 3082	. . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
24	23, 16	wex 1812	. 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 used by:  dcomex  10449  axdc2lem  10450