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Theorem elgch 9645
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 9644 . . . 4 GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)})
21eleq2i 2841 . . 3 (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)}))
3 elun 3902 . . 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 4787 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
6 pweq 4298 . . . . . . . 8 (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
76breq2d 4796 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦𝑥 ≺ 𝒫 𝐴))
85, 7anbi12d 608 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
98notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
109albidv 2000 . . . 4 (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1110elabg 3500 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦𝑥𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1211orbi2d 880 . 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  wa 382  wo 826  wal 1628   = wceq 1630  wcel 2144  {cab 2756  cun 3719  𝒫 cpw 4295   class class class wbr 4784  csdm 8107  Fincfn 8108  GCHcgch 9643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-gch 9644
This theorem is referenced by:  gchi  9647  engch  9651  hargch  9696  alephgch  9697
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