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Theorem gchen1 9444
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 794 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
2 gchi 9443 . . . . . . 7 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323com23 1270 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴𝐴𝐵) → 𝐴 ∈ Fin)
433expia 1266 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) → (𝐴𝐵𝐴 ∈ Fin))
54con3dimp 457 . . . 4 (((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴𝐵)
65an32s 846 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ 𝐴𝐵)
76adantrl 752 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → ¬ 𝐴𝐵)
8 bren2 7983 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
91, 7, 8sylanbrc 698 1 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1989  𝒫 cpw 4156   class class class wbr 4651  cen 7949  cdom 7950  csdm 7951  Fincfn 7952  GCHcgch 9439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-f1o 5893  df-en 7953  df-dom 7954  df-sdom 7955  df-gch 9440
This theorem is referenced by:  gchor  9446  gchcda1  9475  gchcdaidm  9487  gchxpidm  9488  gchhar  9498
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