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Theorem gchi 9005
 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 7525 . . . . . . 7
21brrelexi 5030 . . . . . 6
32adantl 466 . . . . 5
4 breq2 4441 . . . . . . 7
5 breq1 4440 . . . . . . 7
64, 5anbi12d 710 . . . . . 6
76spcegv 3181 . . . . 5
83, 7mpcom 36 . . . 4
9 df-ex 1600 . . . 4
108, 9sylib 196 . . 3
11 elgch 9003 . . . . . 6 GCH GCH
1211ibi 241 . . . . 5 GCH
1312orcomd 388 . . . 4 GCH
1413ord 377 . . 3 GCH
1510, 14syl5 32 . 2 GCH
16153impib 1195 1 GCH
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 368   wa 369   w3a 974  wal 1381   wceq 1383  wex 1599   wcel 1804  cvv 3095  cpw 3997   class class class wbr 4437   csdm 7517  cfn 7518  GCHcgch 9001 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-dom 7520  df-sdom 7521  df-gch 9002 This theorem is referenced by:  gchen1  9006  gchen2  9007  gchpwdom  9051  gchaleph  9052
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