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Theorem elgch 10619
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 variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-gch 10618 . . . 4 GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)})
21eleq2i 2825 . . 3 (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
3 elun 4148 . . 3 (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
42, 3bitri 274 . 2 (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
5 breq1 5151 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
6 pweq 4616 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76breq2d 5160 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦𝑥 ≺ 𝒫 𝐴))
85, 7anbi12d 631 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
98notbid 317 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
109albidv 1923 . . . 4 (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1110elabg 3666 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1211orbi2d 914 . 2 (𝐴𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
134, 12bitrid 282 1 (𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  wal 1539   = wceq 1541  wcel 2106  {cab 2709  cun 3946  𝒫 cpw 4602   class class class wbr 5148  csdm 8940  Fincfn 8941  GCHcgch 10617
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-gch 10618
This theorem is referenced by:  gchi  10621  engch  10625  hargch  10670  alephgch  10671
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