Axiom ax-dc 10551

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 10626. 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 5106	. . . . 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 5656	. . . 4 class ran 𝑥
11	6	cdm 5655	. . . 4 class dom 𝑥
12	10, 11	wss 3900	. . 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 6558	. . . . 5 class (𝑓‘𝑛)
19	15	csuc 6382	. . . . . 6 class suc 𝑛
20	19, 17	cfv 6558	. . . . 5 class (𝑓‘suc 𝑛)
21	18, 20, 6	wbr 5106	. . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)
22		com 7901	. . . 4 class ω
23	21, 14, 22	wral 3080	. . 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  10552  axdc2lem  10553