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Theorem gchen2 10571
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 771 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≼ 𝒫 𝐴)
2 gchi 10569 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323expia 1121 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐴𝐵) → (𝐵 ≺ 𝒫 𝐴𝐴 ∈ Fin))
43con3dimp 409 . . . 4 (((𝐴 ∈ GCH ∧ 𝐴𝐵) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐵 ≺ 𝒫 𝐴)
54an32s 650 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐴𝐵) → ¬ 𝐵 ≺ 𝒫 𝐴)
65adantrr 715 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → ¬ 𝐵 ≺ 𝒫 𝐴)
7 bren2 8930 . 2 (𝐵 ≈ 𝒫 𝐴 ↔ (𝐵 ≼ 𝒫 𝐴 ∧ ¬ 𝐵 ≺ 𝒫 𝐴))
81, 6, 7sylanbrc 583 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2106  𝒫 cpw 4565   class class class wbr 5110  cen 8887  cdom 8888  csdm 8889  Fincfn 8890  GCHcgch 10565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-f1o 6508  df-en 8891  df-dom 8892  df-sdom 8893  df-gch 10566
This theorem is referenced by:  gchhar  10624
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