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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 10496 as our definition, because the equivalence to more standard forms (dfac2 10169) 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 10496 itself as dfac0 10171. (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 10152 | . 2 wff CHOICE | |
2 | vf | . . . . . . 7 setvar 𝑓 | |
3 | 2 | cv 1535 | . . . . . 6 class 𝑓 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1535 | . . . . . 6 class 𝑥 |
6 | 3, 5 | wss 3962 | . . . . 5 wff 𝑓 ⊆ 𝑥 |
7 | 5 | cdm 5688 | . . . . . 6 class dom 𝑥 |
8 | 3, 7 | wfn 6557 | . . . . 5 wff 𝑓 Fn dom 𝑥 |
9 | 6, 8 | wa 395 | . . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
10 | 9, 2 | wex 1775 | . . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
11 | 10, 4 | wal 1534 | . 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 10158 ac7 10510 fineqvac 35089 |
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