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Theorem elgch 9391
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 9390 . . . 4 GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)})
21eleq2i 2690 . . 3 (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
3 elun 3733 . . 3 (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
42, 3bitri 264 . 2 (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
5 breq1 4618 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
6 pweq 4135 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76breq2d 4627 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦𝑥 ≺ 𝒫 𝐴))
85, 7anbi12d 746 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
98notbid 308 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
109albidv 1846 . . . 4 (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1110elabg 3335 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1211orbi2d 737 . 2 (𝐴𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
134, 12syl5bb 272 1 (𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  cun 3554  𝒫 cpw 4132   class class class wbr 4615  csdm 7901  Fincfn 7902  GCHcgch 9389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-gch 9390
This theorem is referenced by:  gchi  9393  engch  9397  hargch  9442  alephgch  9443
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