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Theorem gchen2 10521
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 772 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴)
2 gchi 10519 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323expia 1122 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐴𝐵) → (𝐵 ≺ 𝒫 𝐴𝐴 ∈ Fin))
43con3dimp 410 . . . 4 (((𝐴 ∈ GCH ∧ 𝐴𝐵) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ≺ 𝒫 𝐴)
54an32s 651 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴𝐵) → ¬ 𝐵 ≺ 𝒫 𝐴)
65adantrr 716 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → ¬ 𝐵 ≺ 𝒫 𝐴)
7 bren2 8882 . 2 (𝐵 ≈ 𝒫 𝐴 ↔ (𝐵 ≼ 𝒫 𝐴 ∧ ¬ 𝐵 ≺ 𝒫 𝐴))
81, 6, 7sylanbrc 584 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  𝒫 cpw 4559   class class class wbr 5104  cen 8839  cdom 8840  csdm 8841  Fincfn 8842  GCHcgch 10515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-f1o 6501  df-en 8843  df-dom 8844  df-sdom 8845  df-gch 10516
This theorem is referenced by:  gchhar  10574
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