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Theorem gchen2 9327
 Description: If 𝐴 < 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchen2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)

Proof of Theorem gchen2
StepHypRef Expression
1 simprr 792 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴)
2 gchi 9325 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323expia 1259 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐴𝐵) → (𝐵 ≺ 𝒫 𝐴𝐴 ∈ Fin))
43con3dimp 456 . . . 4 (((𝐴 ∈ GCH ∧ 𝐴𝐵) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ≺ 𝒫 𝐴)
54an32s 842 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴𝐵) → ¬ 𝐵 ≺ 𝒫 𝐴)
65adantrr 749 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → ¬ 𝐵 ≺ 𝒫 𝐴)
7 bren2 7872 . 2 (𝐵 ≈ 𝒫 𝐴 ↔ (𝐵 ≼ 𝒫 𝐴 ∧ ¬ 𝐵 ≺ 𝒫 𝐴))
81, 6, 7sylanbrc 695 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977  𝒫 cpw 4108   class class class wbr 4583   ≈ cen 7838   ≼ cdom 7839   ≺ 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  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-opab 4644  df-xp 5044  df-rel 5045  df-f1o 5811  df-en 7842  df-dom 7843  df-sdom 7844  df-gch 9322 This theorem is referenced by:  gchhar  9380
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