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Theorem eliniseg 5371
Description: Membership in an initial segment. The idiom , meaning , is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
eliniseg.1
Assertion
Ref Expression
eliniseg

Proof of Theorem eliniseg
StepHypRef Expression
1 eliniseg.1 . 2
2 elimasng 5368 . . . 4
3 df-br 4453 . . . 4
42, 3syl6bbr 263 . . 3
5 brcnvg 5188 . . 3
64, 5bitrd 253 . 2
71, 6mpan2 671 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  e.wcel 1818   cvv 3109  {csn 4029  <.cop 4035   class class class wbr 4452  `'ccnv 5003  "cima 5007
This theorem is referenced by:  epini  5372  iniseg  5373  dfco2a  5512  isomin  6233  isoini  6234  fnse  6917  infxpenlem  8412  fpwwe2lem8  9036  fpwwe2lem12  9040  fpwwe2lem13  9041  fpwwe2  9042  canth4  9046  canthwelem  9049  pwfseqlem4  9061  fz1isolem  12510  itg1addlem4  22106  elnlfn  26847  elpred  29257  pw2f1ocnv  30979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017
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