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Theorem dflim3 6682
Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dflim3
Distinct variable group:   ,

Proof of Theorem dflim3
StepHypRef Expression
1 df-lim 4888 . 2
2 3anass 977 . 2
3 df-ne 2654 . . . . . 6
43a1i 11 . . . . 5
5 orduninsuc 6678 . . . . 5
64, 5anbi12d 710 . . . 4
7 ioran 490 . . . 4
86, 7syl6bbr 263 . . 3
98pm5.32i 637 . 2
101, 2, 93bitri 271 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  \/wo 368  /\wa 369  /\w3a 973  =wceq 1395  =/=wne 2652  E.wrex 2808   c0 3784  U.cuni 4249  Ordword 4882   con0 4883  Limwlim 4884  succsuc 4885
This theorem is referenced by:  nlimon  6686  tfinds  6694  oalimcl  7228  omlimcl  7246  r1wunlim  9136
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-8 1820  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  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889
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