MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  acnrcl Structured version   Visualization version   GIF version

Theorem acnrcl 8903
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 3954 . . 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 3987 . . . 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 1898 . . . 4 (∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦)) → 𝐴 ∈ V)
52, 4sylbi 207 . . 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 8806 . 2 AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}
86, 7eleq2s 2748 1 (𝑋AC 𝐴𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  Vcvv 3231  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  AC wacn 8802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-nul 3949  df-acn 8806
This theorem is referenced by:  acni  8906  acni2  8907  acndom2  8915  fodomacn  8917  iundom2g  9400
  Copyright terms: Public domain W3C validator