MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limon Structured version   Unicode version

Theorem limon 4818
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 4765 . 2  |-  Ord  On
2 onn0 4647 . 2  |-  On  =/=  (/)
3 unon 4813 . . 3  |-  U. On  =  On
43eqcomi 2442 . 2  |-  On  =  U. On
5 df-lim 4588 . 2  |-  ( Lim 
On 
<->  ( Ord  On  /\  On  =/=  (/)  /\  On  =  U. On ) )
61, 2, 4, 5mpbir3an 1137 1  |-  Lim  On
Colors of variables: wff set class
Syntax hints:    = wceq 1653    =/= wne 2601   (/)c0 3630   U.cuni 4017   Ord word 4582   Oncon0 4583   Lim wlim 4584
This theorem is referenced by:  limom  4862  oesuc  6773  limensuc  7286  limsucncmp  26198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589
  Copyright terms: Public domain W3C validator