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Theorem gchen1 10035
Description: If 𝐴𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchen1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)

Proof of Theorem gchen1
StepHypRef Expression
1 simprl 767 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
2 gchi 10034 . . . . . . 7 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323com23 1118 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴𝐴𝐵) → 𝐴 ∈ Fin)
433expia 1113 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) → (𝐴𝐵𝐴 ∈ Fin))
54con3dimp 409 . . . 4 (((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴𝐵)
65an32s 648 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ 𝐴𝐵)
76adantrl 712 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → ¬ 𝐴𝐵)
8 bren2 8528 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
91, 7, 8sylanbrc 583 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2105  𝒫 cpw 4535   class class class wbr 5057  cen 8494  cdom 8495  csdm 8496  Fincfn 8497  GCHcgch 10030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-f1o 6355  df-en 8498  df-dom 8499  df-sdom 8500  df-gch 10031
This theorem is referenced by:  gchor  10037  gchdju1  10066  gchdjuidm  10078  gchxpidm  10079  gchhar  10089
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