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

Theorem gchi 10664
Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gchi ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)

Proof of Theorem gchi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relsdom 8992 . . . . . . 7 Rel ≺
21brrelex1i 5741 . . . . . 6 (𝐵 ≺ 𝒫 𝐴𝐵 ∈ V)
32adantl 481 . . . . 5 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4 breq2 5147 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
5 breq1 5146 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴𝐵 ≺ 𝒫 𝐴))
64, 5anbi12d 632 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)))
76spcegv 3597 . . . . 5 (𝐵 ∈ V → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
83, 7mpcom 38 . . . 4 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴))
9 df-ex 1780 . . . 4 (∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
108, 9sylib 218 . . 3 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
11 elgch 10662 . . . . . 6 (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
1211ibi 267 . . . . 5 (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1312orcomd 872 . . . 4 (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
1413ord 865 . . 3 (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
1510, 14syl5 34 . 2 (𝐴 ∈ GCH → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
16153impib 1117 1 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143  csdm 8984  Fincfn 8985  GCHcgch 10660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dom 8987  df-sdom 8988  df-gch 10661
This theorem is referenced by:  gchen1  10665  gchen2  10666  gchpwdom  10710  gchaleph  10711
  Copyright terms: Public domain W3C validator