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Theorem acnrcl 10067
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 4334 . . 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 4382 . . . 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 481 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) → 𝐴 ∈ V)
43exlimiv 1925 . . . 4 (∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) → 𝐴 ∈ V)
52, 4sylbi 216 . . 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 9967 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
86, 7eleq2s 2843 1 (𝑋AC 𝐴𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1773  wcel 2098  {cab 2702  wne 2929  wral 3050  Vcvv 3461  cdif 3941  c0 4322  𝒫 cpw 4604  {csn 4630  cfv 6549  (class class class)co 7419  m cmap 8845  AC wacn 9963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-dif 3947  df-nul 4323  df-acn 9967
This theorem is referenced by:  acni  10070  acni2  10071  acndom2  10079  fodomacn  10081  iundom2g  10565
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