Theorem acnrcl 7391

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 𝑧 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2816	. . 3 ⊢ (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} → ∃𝑤 𝑤 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))})
2		abn0m 3517	. . . 4 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} ↔ ∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)))
3		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → 𝐴 ∈ V)
4	3	exlimiv 1644	. . . 4 ⊢ (∃𝑥(𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → 𝐴 ∈ V)
5	2, 4	sylbi 121	. . 3 ⊢ (∃𝑤 𝑤 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} → 𝐴 ∈ V)
6	1, 5	syl 14	. 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} → 𝐴 ∈ V)
7		df-acnm 7360	. 2 ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
8	6, 7	eleq2s 2324	1 ⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∀wral 2508  {crab 2512  Vcvv 2799  𝒫 cpw 3649  ‘cfv 5318  (class class class)co 6007   ↑_𝑚 cmap 6803  AC wacn 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801  df-acnm 7360

This theorem is referenced by: (None)