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Theorem elgch 9323
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 9322 . . . 4 GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)})
21eleq2i 2680 . . 3 (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
3 elun 3715 . . 3 (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
42, 3bitri 263 . 2 (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
5 breq1 4586 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
6 pweq 4111 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76breq2d 4595 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦𝑥 ≺ 𝒫 𝐴))
85, 7anbi12d 743 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
98notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
109albidv 1836 . . . 4 (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1110elabg 3320 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1211orbi2d 734 . 2 (𝐴𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
134, 12syl5bb 271 1 (𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  wal 1473   = wceq 1475  wcel 1977  {cab 2596  cun 3538  𝒫 cpw 4108   class class class wbr 4583  csdm 7840  Fincfn 7841  GCHcgch 9321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-gch 9322
This theorem is referenced by:  gchi  9325  engch  9329  hargch  9374  alephgch  9375
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