Theorem acnrcl 8865

Description: Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acnrcl	⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)

Proof of Theorem acnrcl

Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ne0i 3921	. . 3 ⊢ (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} → {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} ≠ ∅)
2		abn0 3954	. . . 4 ⊢ ({𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} ≠ ∅ ↔ ∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
3		simpl 473	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → 𝐴 ∈ V)
4	3	exlimiv 1858	. . . 4 ⊢ (∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → 𝐴 ∈ V)
5	2, 4	sylbi 207	. . 3 ⊢ ({𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} ≠ ∅ → 𝐴 ∈ V)
6	1, 5	syl 17	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} → 𝐴 ∈ V)
7		df-acn 8768	. 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
8	6, 7	eleq2s 2719	1 ⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  AC wacn 8764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916  df-acn 8768

This theorem is referenced by:  acni  8868  acni2  8869  acndom2  8877  fodomacn  8879  iundom2g  9362