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Theorem gchi 10038
 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 8502 . . . . . . 7 Rel ≺
21brrelex1i 5573 . . . . . 6 (𝐵 ≺ 𝒫 𝐴𝐵 ∈ V)
32adantl 485 . . . . 5 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4 breq2 5035 . . . . . . 7 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
5 breq1 5034 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴𝐵 ≺ 𝒫 𝐴))
64, 5anbi12d 633 . . . . . 6 (𝑥 = 𝐵 → ((𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)))
76spcegv 3545 . . . . 5 (𝐵 ∈ V → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
83, 7mpcom 38 . . . 4 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴))
9 df-ex 1782 . . . 4 (∃𝑥(𝐴𝑥𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
108, 9sylib 221 . . 3 ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
11 elgch 10036 . . . . . 6 (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
1211ibi 270 . . . . 5 (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
1312orcomd 868 . . . 4 (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
1413ord 861 . . 3 (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
1510, 14syl5 34 . 2 (𝐴 ∈ GCH → ((𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
16153impib 1113 1 ((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5031   ≺ csdm 8494  Fincfn 8495  GCHcgch 10034 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-dom 8497  df-sdom 8498  df-gch 10035 This theorem is referenced by:  gchen1  10039  gchen2  10040  gchpwdom  10084  gchaleph  10085
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