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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 9164 as our definition, because the equivalence to more standard forms (dfac2 8836) 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 9164 itself as dfac0 8838. (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 8821 | . 2 wff CHOICE | |
2 | vf | . . . . . . 7 setvar 𝑓 | |
3 | 2 | cv 1474 | . . . . . 6 class 𝑓 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 4 | cv 1474 | . . . . . 6 class 𝑥 |
6 | 3, 5 | wss 3540 | . . . . 5 wff 𝑓 ⊆ 𝑥 |
7 | 5 | cdm 5038 | . . . . . 6 class dom 𝑥 |
8 | 3, 7 | wfn 5799 | . . . . 5 wff 𝑓 Fn dom 𝑥 |
9 | 6, 8 | wa 383 | . . . 4 wff (𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
10 | 9, 2 | wex 1695 | . . 3 wff ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
11 | 10, 4 | wal 1473 | . 2 wff ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) |
12 | 1, 11 | wb 195 | 1 wff (CHOICE ↔ ∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: dfac3 8827 ac7 9178 |
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