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Theorem acnrcl 8748
 Description: Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acnrcl (𝑋AC 𝐴𝐴 ∈ V)

Proof of Theorem acnrcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3880 . . 3 (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} → {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} ≠ ∅)
2 abn0 3908 . . . 4 ({𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} ≠ ∅ ↔ ∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)))
3 simpl 472 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) → 𝐴 ∈ V)
43exlimiv 1845 . . . 4 (∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) → 𝐴 ∈ V)
52, 4sylbi 206 . . 3 ({𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} ≠ ∅ → 𝐴 ∈ V)
61, 5syl 17 . 2 (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))} → 𝐴 ∈ V)
7 df-acn 8651 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
86, 7eleq2s 2706 1 (𝑋AC 𝐴𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  AC wacn 8647 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875  df-acn 8651 This theorem is referenced by:  acni  8751  acni2  8752  acndom2  8760  fodomacn  8762  iundom2g  9241
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