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Theorem nlimon 4822
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 4583 . . 3  |-  ( x  e.  On  ->  Ord  x )
2 dflim3 4818 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
32baib 872 . . . 4  |-  ( Ord  x  ->  ( Lim  x 
<->  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43con2bid 320 . . 3  |-  ( Ord  x  ->  ( (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x
) )
51, 4syl 16 . 2  |-  ( x  e.  On  ->  (
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x ) )
65rabbiia 2938 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 177    \/ wo 358    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   (/)c0 3620   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
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