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Theorem dfac8c 9056
 Description: If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
Assertion
Ref Expression
dfac8c (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Distinct variable groups:   𝑓,𝑟,𝑧,𝐴   𝐵,𝑟
Allowed substitution hints:   𝐵(𝑧,𝑓)

Proof of Theorem dfac8c
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . 2 (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (𝑦𝑥𝑤𝑥 ¬ 𝑤𝑟𝑦)) = (𝑥 ∈ (𝐴 ∖ {∅}) ↦ (𝑦𝑥𝑤𝑥 ¬ 𝑤𝑟𝑦))
21dfac8clem 9055 1 (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061   ∖ cdif 3720  ∅c0 4063  {csn 4316  ∪ cuni 4574   class class class wbr 4786   ↦ cmpt 4863   We wwe 5207  ‘cfv 6031  ℩crio 6753 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-riota 6754 This theorem is referenced by:  ween  9058  ac5num  9059  dfac8  9159  vitali  23601
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