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Theorem limon 4807
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 4754 . 2  |-  Ord  On
2 onn0 4637 . 2  |-  On  =/=  (/)
3 unon 4802 . . 3  |-  U. On  =  On
43eqcomi 2439 . 2  |-  On  =  U. On
5 df-lim 4578 . 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 1652    =/= wne 2598   (/)c0 3620   U.cuni 4007   Ord word 4572   Oncon0 4573   Lim wlim 4574
This theorem is referenced by:  limom  4851  oesuc  6762  limensuc  7275  limsucncmp  26144
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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