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| 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 10561. 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 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 5143 | . . . . 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 5686 | . . . 4 class ran 𝑥 |
| 11 | 6 | cdm 5685 | . . . 4 class dom 𝑥 |
| 12 | 10, 11 | wss 3951 | . . 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 6561 | . . . . 5 class (𝑓‘𝑛) |
| 19 | 15 | csuc 6386 | . . . . . 6 class suc 𝑛 |
| 20 | 19, 17 | cfv 6561 | . . . . 5 class (𝑓‘suc 𝑛) |
| 21 | 18, 20, 6 | wbr 5143 | . . . 4 wff (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) |
| 22 | com 7887 | . . . 4 class ω | |
| 23 | 21, 14, 22 | wral 3061 | . . 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)) |
| Colors of variables: wff setvar class |
| This axiom is referenced by: dcomex 10487 axdc2lem 10488 |
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