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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 9881 as our definition, because the equivalence to more standard forms (dfac2 9557) 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 9881 itself as dfac0 9559. (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 9541 | . 2 wff CHOICE | |
2 | vf | . . . . . . 7 setvar 𝑓 | |
3 | 2 | cv 1536 | . . . . . 6 class 𝑓 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1536 | . . . . . 6 class 𝑥 |
6 | 3, 5 | wss 3936 | . . . . 5 wff 𝑓 ⊆ 𝑥 |
7 | 5 | cdm 5555 | . . . . . 6 class dom 𝑥 |
8 | 3, 7 | wfn 6350 | . . . . 5 wff 𝑓 Fn dom 𝑥 |
9 | 6, 8 | wa 398 | . . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
10 | 9, 2 | wex 1780 | . . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
11 | 10, 4 | wal 1535 | . 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
12 | 1, 11 | wb 208 | 1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfac3 9547 ac7 9895 |
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