MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gchen1 Structured version   Visualization version   GIF version

Theorem gchen1 10663
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 771 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
2 gchi 10662 . . . . . . 7 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
323com23 1125 . . . . . 6 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴𝐴𝐵) → 𝐴 ∈ Fin)
433expia 1120 . . . . 5 ((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) → (𝐴𝐵𝐴 ∈ Fin))
54con3dimp 408 . . . 4 (((𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴) ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴𝐵)
65an32s 652 . . 3 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ 𝐴𝐵)
76adantrl 716 . 2 (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → ¬ 𝐴𝐵)
8 bren2 9022 . 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 395  wcel 2106  𝒫 cpw 4605   class class class wbr 5148  cen 8981  cdom 8982  csdm 8983  Fincfn 8984  GCHcgch 10658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987  df-gch 10659
This theorem is referenced by:  gchor  10665  gchdju1  10694  gchdjuidm  10706  gchxpidm  10707  gchhar  10717
  Copyright terms: Public domain W3C validator