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Theorem gchi 8491
 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 GCH

Proof of Theorem gchi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 relsdom 7108 . . . . . . 7
21brrelexi 4910 . . . . . 6
32adantl 453 . . . . 5
4 breq2 4208 . . . . . . 7
5 breq1 4207 . . . . . . 7
64, 5anbi12d 692 . . . . . 6
76spcegv 3029 . . . . 5
83, 7mpcom 34 . . . 4
9 df-ex 1551 . . . 4
108, 9sylib 189 . . 3
11 elgch 8489 . . . . . 6 GCH GCH
1211ibi 233 . . . . 5 GCH
1312orcomd 378 . . . 4 GCH
1413ord 367 . . 3 GCH
1510, 14syl5 30 . 2 GCH
16153impib 1151 1 GCH
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wa 359   w3a 936  wal 1549  wex 1550   wceq 1652   wcel 1725  cvv 2948  cpw 3791   class class class wbr 4204   csdm 7100  cfn 7101  GCHcgch 8487 This theorem is referenced by:  gchen1  8492  gchen2  8493  gchpwdom  8541  gchaleph  8542 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-dom 7103  df-sdom 7104  df-gch 8488
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