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Mirrors > Home > ILE Home > Th. List > df-ac | 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 are some decisions about how to write this definition especially around whether ax-setind 4498 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (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 7142 | . 2 wff CHOICE | |
2 | vf | . . . . . . 7 setvar 𝑓 | |
3 | 2 | cv 1334 | . . . . . 6 class 𝑓 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1334 | . . . . . 6 class 𝑥 |
6 | 3, 5 | wss 3102 | . . . . 5 wff 𝑓 ⊆ 𝑥 |
7 | 5 | cdm 4588 | . . . . . 6 class dom 𝑥 |
8 | 3, 7 | wfn 5167 | . . . . 5 wff 𝑓 Fn dom 𝑥 |
9 | 6, 8 | wa 103 | . . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
10 | 9, 2 | wex 1472 | . . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
11 | 10, 4 | wal 1333 | . 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
12 | 1, 11 | wb 104 | 1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
Colors of variables: wff set class |
This definition is referenced by: acfun 7144 |
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