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Theorem nlimon 4614
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 4374 . . 3  |-  ( x  e.  On  ->  Ord  x )
2 dflim3 4610 . . . . 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 2753 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 1619    e. wcel 1621   E.wrex 2519   {crab 2522   (/)c0 3430   Ord word 4363   Oncon0 4364   Lim wlim 4365   suc csuc 4366
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370
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