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| Mirrors > Home > MPE Home > Th. List > df-ac | Structured version Visualization version GIF version | ||
| Description: The expression CHOICE will be used as a readable shorthand
for any
form of the axiom of choice; all concrete forms are long, cryptic, have
dummy variables, or all three, making it useful to have a short name.
Similar to the Axiom of Choice (first form) of [Enderton] p. 49.
There is a slight problem with taking the exact form of ax-ac 10499 as our definition, because the equivalence to more standard forms (dfac2 10172) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 10499 itself as dfac0 10174. (Contributed by Mario Carneiro, 22-Feb-2015.) |
| Ref | Expression |
|---|---|
| df-ac | ⊢ (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wac 10155 | . 2 wff CHOICE | |
| 2 | vf | . . . . . . 7 setvar 𝑓 | |
| 3 | 2 | cv 1539 | . . . . . 6 class 𝑓 |
| 4 | vx | . . . . . . 7 setvar 𝑥 | |
| 5 | 4 | cv 1539 | . . . . . 6 class 𝑥 |
| 6 | 3, 5 | wss 3951 | . . . . 5 wff 𝑓 ⊆ 𝑥 |
| 7 | 5 | cdm 5685 | . . . . . 6 class dom 𝑥 |
| 8 | 3, 7 | wfn 6556 | . . . . 5 wff 𝑓 Fn dom 𝑥 |
| 9 | 6, 8 | wa 395 | . . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
| 10 | 9, 2 | wex 1779 | . . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
| 11 | 10, 4 | wal 1538 | . 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
| 12 | 1, 11 | wb 206 | 1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
| Colors of variables: wff setvar class |
| This definition is referenced by: dfac3 10161 ac7 10513 fineqvac 35111 |
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