MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlimon Unicode version

Theorem nlimon 4642
Description: Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.)
Assertion
Ref Expression
nlimon  |-  { x  e.  On  |  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) }  =  { x  e.  On  |  -.  Lim  x }
Distinct variable group:    x, y

Proof of Theorem nlimon
StepHypRef Expression
1 eloni 4402 . . 3  |-  ( x  e.  On  ->  Ord  x )
2 dflim3 4638 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
32baib 871 . . . 4  |-  ( Ord  x  ->  ( Lim  x 
<->  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43con2bid 319 . . 3  |-  ( Ord  x  ->  ( (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x
) )
51, 4syl 15 . 2  |-  ( x  e.  On  ->  (
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x ) )
65rabbiia 2778 1  |-  { x  e.  On  |  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) }  =  { x  e.  On  |  -.  Lim  x }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   (/)c0 3455   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
  Copyright terms: Public domain W3C validator