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Theorem nlimon 6565
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 4830 . . 3  |-  ( x  e.  On  ->  Ord  x )
2 dflim3 6561 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
32baib 896 . . . 4  |-  ( Ord  x  ->  ( Lim  x 
<->  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43con2bid 329 . . 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 3060 1  |-  { x  e.  On  |  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) }  =  { x  e.  On  |  -.  Lim  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758   E.wrex 2796   {crab 2799   (/)c0 3738   Ord word 4819   Oncon0 4820   Lim wlim 4821   suc csuc 4822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826
This theorem is referenced by: (None)
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