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Theorem limon 7078
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 7024 . 2 Ord On
2 onn0 5827 . 2 On ≠ ∅
3 unon 7073 . . 3 On = On
43eqcomi 2660 . 2 On = On
5 df-lim 5766 . 2 (Lim On ↔ (Ord On ∧ On ≠ ∅ ∧ On = On))
61, 2, 4, 5mpbir3an 1263 1 Lim On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wne 2823  c0 3948   cuni 4468  Ord word 5760  Oncon0 5761  Lim wlim 5762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767
This theorem is referenced by:  limom  7122  oesuc  7652  limensuc  8178  limsucncmp  32570
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