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Theorem ac3 9027
Description: Axiom of Choice using abbreviations. The logical equivalence to ax-ac 9024 can be established by chaining aceq0 8684 and aceq2 8685. A standard textbook version of AC is derived from this one in dfac2a 8695, and this version of AC is derived from the textbook version in dfac2 8696.

The following sketch will help you understand this version of the axiom. Given any set 𝑥, the axiom says that 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. Using the Axiom of Regularity, we can show that 𝑦 is really a set of ordered pairs, very similar to the ordered pair construction opthreg 8258. The key theorem for this (used in the proof of dfac2 8696) is preleq 8257. With this modified definition of ordered pair, it can be seen that 𝑦 is actually a choice function on the members of 𝑥.

For example, suppose 𝑥 = {{1, 2}, {1, 3}, {2, 3, 4}}. Let us try 𝑦 = {{{1, 2}, 1}, {{1, 3}, 1}, {{2, 3, 4}, 2}}. For the member (of 𝑥) 𝑧 = {1, 2}, the only assignment to 𝑤 and 𝑣 that satisfies the axiom is 𝑤 = 1 and 𝑣 = {{1, 2}, 1}, so there is exactly one 𝑤 as required. We verify the other two members of 𝑥 similarly. Thus, 𝑦 satisfies the axiom. Using our modified ordered pair definition, we can say that 𝑦 corresponds to the choice function {⟨{1, 2}, 1⟩, ⟨{1, 3}, 1⟩, ⟨{2, 3, 4}, 2⟩}. Of course other choices for 𝑦 will also satisfy the axiom, for example 𝑦 = {{{1, 2}, 2}, {{1, 3}, 1}, {{2, 3, 4}, 4}}. What AC tells us is that there exists at least one such 𝑦, but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

Assertion
Ref Expression
ac3 𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣

Proof of Theorem ac3
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ac2 9026 . 2 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
2 aceq2 8685 . 2 (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)))
31, 2mpbi 218 1 𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 381  wex 1694  wne 2687  wral 2803  wrex 2804  ∃!wreu 2805  c0 3779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1941  ax-9 1948  ax-10 1967  ax-11 1972  ax-12 1985  ax-13 2140  ax-ext 2497  ax-ac 9024
This theorem depends on definitions:  df-bi 195  df-or 382  df-an 383  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2369  df-clab 2504  df-cleq 2510  df-clel 2513  df-nfc 2647  df-ne 2689  df-ral 2808  df-rex 2809  df-reu 2810  df-v 3082  df-dif 3449  df-nul 3780
This theorem is referenced by:  axac2  9031
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