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Theorem elgch 10037
 Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
elgch (𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elgch
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-gch 10036 . . . 4 GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)})
21eleq2i 2884 . . 3 (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
3 elun 4079 . . 3 (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
42, 3bitri 278 . 2 (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
5 breq1 5036 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝑥𝑧𝑥))
6 pweq 4516 . . . . . . . 8 (𝑦 = 𝑧 → 𝒫 𝑦 = 𝒫 𝑧)
76breq2d 5045 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 ≺ 𝒫 𝑦𝑥 ≺ 𝒫 𝑧))
85, 7anbi12d 633 . . . . . 6 (𝑦 = 𝑧 → ((𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ (𝑧𝑥𝑥 ≺ 𝒫 𝑧)))
98notbid 321 . . . . 5 (𝑦 = 𝑧 → (¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝑧𝑥𝑥 ≺ 𝒫 𝑧)))
109albidv 1921 . . . 4 (𝑦 = 𝑧 → (∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝑧𝑥𝑥 ≺ 𝒫 𝑧)))
11 breq1 5036 . . . . . . 7 (𝑧 = 𝐴 → (𝑧𝑥𝐴𝑥))
12 pweq 4516 . . . . . . . 8 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
1312breq2d 5045 . . . . . . 7 (𝑧 = 𝐴 → (𝑥 ≺ 𝒫 𝑧𝑥 ≺ 𝒫 𝐴))
1411, 13anbi12d 633 . . . . . 6 (𝑧 = 𝐴 → ((𝑧𝑥𝑥 ≺ 𝒫 𝑧) ↔ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1514notbid 321 . . . . 5 (𝑧 = 𝐴 → (¬ (𝑧𝑥𝑥 ≺ 𝒫 𝑧) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1615albidv 1921 . . . 4 (𝑧 = 𝐴 → (∀𝑥 ¬ (𝑧𝑥𝑥 ≺ 𝒫 𝑧) ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1710, 16elabgw 3615 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1817orbi2d 913 . 2 (𝐴𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
194, 18syl5bb 286 1 (𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2112  {cab 2779   ∪ cun 3882  𝒫 cpw 4500   class class class wbr 5033   ≺ csdm 8495  Fincfn 8496  GCHcgch 10035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-gch 10036 This theorem is referenced by:  gchi  10039  engch  10043  hargch  10088  alephgch  10089
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