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Theorem limon 4757
Description: The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Assertion
Ref Expression
limon  |-  Lim  On

Proof of Theorem limon
StepHypRef Expression
1 ordon 4704 . 2  |-  Ord  On
2 onn0 4587 . 2  |-  On  =/=  (/)
3 unon 4752 . . 3  |-  U. On  =  On
43eqcomi 2392 . 2  |-  On  =  U. On
5 df-lim 4528 . 2  |-  ( Lim 
On 
<->  ( Ord  On  /\  On  =/=  (/)  /\  On  =  U. On ) )
61, 2, 4, 5mpbir3an 1136 1  |-  Lim  On
Colors of variables: wff set class
Syntax hints:    = wceq 1649    =/= wne 2551   (/)c0 3572   U.cuni 3958   Ord word 4522   Oncon0 4523   Lim wlim 4524
This theorem is referenced by:  limom  4801  oesuc  6708  limensuc  7221  limsucncmp  25911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529
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