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Theorem gchi 10619
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 8946 . . . . . . 7 Rel ≺
21brrelex1i 5733 . . . . . 6 (𝐵 ≺ 𝒫 𝐴𝐵 ∈ V)
32adantl 483 . . . . 5 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4 breq2 5153 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
5 breq1 5152 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴𝐵 ≺ 𝒫 𝐴))
64, 5anbi12d 632 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)))
76spcegv 3588 . . . . 5 (𝐵 ∈ V → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
83, 7mpcom 38 . . . 4 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴))
9 df-ex 1783 . . . 4 (∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
108, 9sylib 217 . . 3 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
11 elgch 10617 . . . . . 6 (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
1211ibi 267 . . . . 5 (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1312orcomd 870 . . . 4 (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
1413ord 863 . . 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 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  𝒫 cpw 4603   class class class wbr 5149  csdm 8938  Fincfn 8939  GCHcgch 10615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dom 8941  df-sdom 8942  df-gch 10616
This theorem is referenced by:  gchen1  10620  gchen2  10621  gchpwdom  10665  gchaleph  10666
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