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Mirrors > Home > MPE Home > Th. List > ax-ac | Structured version Visualization version GIF version |
Description: Axiom of Choice. The
Axiom of Choice (AC) is usually considered an
extension of ZF set theory rather than a proper part of it. It is
sometimes considered philosophically controversial because it asserts
the existence of a set without telling us what the set is. ZF set
theory that includes AC is called ZFC.
The unpublished version given here says that given any set 𝑥, there exists a 𝑦 that is a collection of unordered pairs, one pair for each nonempty member of 𝑥. One entry in the pair is the member of 𝑥, and the other entry is some arbitrary member of that member of 𝑥. See the rewritten version ac3 9167 for a more detailed explanation. Theorem ac2 9166 shows an equivalent written compactly with restricted quantifiers. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 9170 is slightly shorter when the biconditional of ax-ac 9164 is expanded into implication and negation. In axac3 9169 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9382 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 9197, ac5 9182, and ac7 9178. The Axiom of Regularity ax-reg 8380 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8836. Equivalents to AC are the well-ordering theorem weth 9200 and Zorn's lemma zorn 9212. See ac4 9180 for comments about stronger versions of AC. In order to avoid uses of ax-reg 8380 for derivation of AC equivalents, we provide ax-ac2 9168 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 9168 from ax-ac 9164 is shown by theorem axac2 9171, and the reverse derivation by axac 9172. Therefore, new proofs should normally use ax-ac2 9168 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.) |
Ref | Expression |
---|---|
ax-ac | ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vz | . . . . . . 7 setvar 𝑧 | |
2 | vw | . . . . . . 7 setvar 𝑤 | |
3 | 1, 2 | wel 1978 | . . . . . 6 wff 𝑧 ∈ 𝑤 |
4 | vx | . . . . . . 7 setvar 𝑥 | |
5 | 2, 4 | wel 1978 | . . . . . 6 wff 𝑤 ∈ 𝑥 |
6 | 3, 5 | wa 383 | . . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) |
7 | vu | . . . . . . . . . . . 12 setvar 𝑢 | |
8 | 7, 2 | wel 1978 | . . . . . . . . . . 11 wff 𝑢 ∈ 𝑤 |
9 | vt | . . . . . . . . . . . 12 setvar 𝑡 | |
10 | 2, 9 | wel 1978 | . . . . . . . . . . 11 wff 𝑤 ∈ 𝑡 |
11 | 8, 10 | wa 383 | . . . . . . . . . 10 wff (𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) |
12 | 7, 9 | wel 1978 | . . . . . . . . . . 11 wff 𝑢 ∈ 𝑡 |
13 | vy | . . . . . . . . . . . 12 setvar 𝑦 | |
14 | 9, 13 | wel 1978 | . . . . . . . . . . 11 wff 𝑡 ∈ 𝑦 |
15 | 12, 14 | wa 383 | . . . . . . . . . 10 wff (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦) |
16 | 11, 15 | wa 383 | . . . . . . . . 9 wff ((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) |
17 | 16, 9 | wex 1695 | . . . . . . . 8 wff ∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) |
18 | vv | . . . . . . . . 9 setvar 𝑣 | |
19 | 7, 18 | weq 1861 | . . . . . . . 8 wff 𝑢 = 𝑣 |
20 | 17, 19 | wb 195 | . . . . . . 7 wff (∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) |
21 | 20, 7 | wal 1473 | . . . . . 6 wff ∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) |
22 | 21, 18 | wex 1695 | . . . . 5 wff ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣) |
23 | 6, 22 | wi 4 | . . . 4 wff ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
24 | 23, 2 | wal 1473 | . . 3 wff ∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
25 | 24, 1 | wal 1473 | . 2 wff ∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
26 | 25, 13 | wex 1695 | 1 wff ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
Colors of variables: wff setvar class |
This axiom is referenced by: zfac 9165 ac2 9166 |
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