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Theorem elgch 8996
 Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
elgch GCH
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem elgch
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-gch 8995 . . . 4 GCH
21eleq2i 2545 . . 3 GCH
3 elun 3645 . . 3
42, 3bitri 249 . 2 GCH
5 breq1 4450 . . . . . . 7
6 pweq 4013 . . . . . . . 8
76breq2d 4459 . . . . . . 7
85, 7anbi12d 710 . . . . . 6
98notbid 294 . . . . 5
109albidv 1689 . . . 4
1110elabg 3251 . . 3
1211orbi2d 701 . 2
134, 12syl5bb 257 1 GCH
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wo 368   wa 369  wal 1377   wceq 1379   wcel 1767  cab 2452   cun 3474  cpw 4010   class class class wbr 4447   csdm 7512  cfn 7513  GCHcgch 8994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-gch 8995 This theorem is referenced by:  gchi  8998  engch  9002  hargch  9047  alephgch  9048
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