![]() |
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 10558. 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 1535 | . . . . . 6 class 𝑦 |
3 | vz | . . . . . . 7 setvar 𝑧 | |
4 | 3 | cv 1535 | . . . . . 6 class 𝑧 |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1535 | . . . . . 6 class 𝑥 |
7 | 2, 4, 6 | wbr 5147 | . . . . 5 wff 𝑦𝑥𝑧 |
8 | 7, 3 | wex 1775 | . . . 4 wff ∃𝑧 𝑦𝑥𝑧 |
9 | 8, 1 | wex 1775 | . . 3 wff ∃𝑦∃𝑧 𝑦𝑥𝑧 |
10 | 6 | crn 5689 | . . . 4 class ran 𝑥 |
11 | 6 | cdm 5688 | . . . 4 class dom 𝑥 |
12 | 10, 11 | wss 3962 | . . 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 1535 | . . . . . 6 class 𝑛 |
16 | vf | . . . . . . 7 setvar 𝑓 | |
17 | 16 | cv 1535 | . . . . . 6 class 𝑓 |
18 | 15, 17 | cfv 6562 | . . . . 5 class (𝑓‘𝑛) |
19 | 15 | csuc 6387 | . . . . . 6 class suc 𝑛 |
20 | 19, 17 | cfv 6562 | . . . . 5 class (𝑓‘suc 𝑛) |
21 | 18, 20, 6 | wbr 5147 | . . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) |
22 | com 7886 | . . . 4 class ω | |
23 | 21, 14, 22 | wral 3058 | . . 3 wff ∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) |
24 | 23, 16 | wex 1775 | . 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 10484 axdc2lem 10485 |
Copyright terms: Public domain | W3C validator |