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Theorem gchi 10034
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 8504 . . . . . . 7 Rel ≺
21brrelex1i 5601 . . . . . 6 (𝐵 ≺ 𝒫 𝐴𝐵 ∈ V)
32adantl 482 . . . . 5 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4 breq2 5061 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
5 breq1 5060 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴𝐵 ≺ 𝒫 𝐴))
64, 5anbi12d 630 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)))
76spcegv 3594 . . . . 5 (𝐵 ∈ V → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
83, 7mpcom 38 . . . 4 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴))
9 df-ex 1772 . . . 4 (∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
108, 9sylib 219 . . 3 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
11 elgch 10032 . . . . . 6 (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
1211ibi 268 . . . . 5 (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1312orcomd 865 . . . 4 (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
1413ord 858 . . 3 (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
1510, 14syl5 34 . 2 (𝐴 ∈ GCH → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
16153impib 1108 1 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  𝒫 cpw 4535   class class class wbr 5057  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-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-dom 8499  df-sdom 8500  df-gch 10031
This theorem is referenced by:  gchen1  10035  gchen2  10036  gchpwdom  10080  gchaleph  10081
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