![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ax-dc | Structured version Visualization version GIF version |
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 9680. 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.) |
Ref | Expression |
---|---|
ax-dc | ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vy | . . . . . . 7 setvar 𝑦 | |
2 | 1 | cv 1600 | . . . . . 6 class 𝑦 |
3 | vz | . . . . . . 7 setvar 𝑧 | |
4 | 3 | cv 1600 | . . . . . 6 class 𝑧 |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1600 | . . . . . 6 class 𝑥 |
7 | 2, 4, 6 | wbr 4888 | . . . . 5 wff 𝑦𝑥𝑧 |
8 | 7, 3 | wex 1823 | . . . 4 wff ∃𝑧 𝑦𝑥𝑧 |
9 | 8, 1 | wex 1823 | . . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧 |
10 | 6 | crn 5358 | . . . 4 class ran 𝑥 |
11 | 6 | cdm 5357 | . . . 4 class dom 𝑥 |
12 | 10, 11 | wss 3792 | . . 3 wff ran 𝑥 ⊆ dom 𝑥 |
13 | 9, 12 | wa 386 | . 2 wff (∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) |
14 | vn | . . . . . . 7 setvar 𝑛 | |
15 | 14 | cv 1600 | . . . . . 6 class 𝑛 |
16 | vf | . . . . . . 7 setvar 𝑓 | |
17 | 16 | cv 1600 | . . . . . 6 class 𝑓 |
18 | 15, 17 | cfv 6137 | . . . . 5 class (𝑓‘𝑛) |
19 | 15 | csuc 5980 | . . . . . 6 class suc 𝑛 |
20 | 19, 17 | cfv 6137 | . . . . 5 class (𝑓‘suc 𝑛) |
21 | 18, 20, 6 | wbr 4888 | . . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) |
22 | com 7345 | . . . 4 class ω | |
23 | 21, 14, 22 | wral 3090 | . . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) |
24 | 23, 16 | wex 1823 | . 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)) |
Colors of variables: wff setvar class |
This axiom is referenced by: dcomex 9606 axdc2lem 9607 |
Copyright terms: Public domain | W3C validator |