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Theorem gchi 10662
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 8991 . . . . . . 7 Rel ≺
21brrelex1i 5745 . . . . . 6 (𝐵 ≺ 𝒫 𝐴𝐵 ∈ V)
32adantl 481 . . . . 5 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4 breq2 5152 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
5 breq1 5151 . . . . . . 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 1777 . . . 4 (∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
108, 9sylib 218 . . 3 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
11 elgch 10660 . . . . . 6 (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
1211ibi 267 . . . . 5 (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1312orcomd 871 . . . 4 (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
1413ord 864 . . 3 (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
1510, 14syl5 34 . 2 (𝐴 ∈ GCH → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
16153impib 1115 1 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  𝒫 cpw 4605   class class class wbr 5148  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-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-sb 2063  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-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-dom 8986  df-sdom 8987  df-gch 10659
This theorem is referenced by:  gchen1  10663  gchen2  10664  gchpwdom  10708  gchaleph  10709
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