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Theorem nlimon 4641
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 4401 . . 3  |-  ( x  e.  On  ->  Ord  x )
2 dflim3 4637 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
32baib 876 . . . 4  |-  ( Ord  x  ->  ( Lim  x 
<->  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43con2bid 321 . . 3  |-  ( Ord  x  ->  ( (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x
) )
51, 4syl 17 . 2  |-  ( x  e.  On  ->  (
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y )  <->  -.  Lim  x ) )
65rabbiia 2779 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 5    <-> wb 178    \/ wo 359    = wceq 1628    e. wcel 1688   E.wrex 2545   {crab 2548   (/)c0 3456   Ord word 4390   Oncon0 4391   Lim wlim 4392   suc csuc 4393
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397
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