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Theorem limon 4514
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 4487 . 2 Ord On
2 0elon 4394 . 2 ∅ ∈ On
3 unon 4512 . . 3 On = On
43eqcomi 2181 . 2 On = On
5 dflim2 4372 . 2 (Lim On ↔ (Ord On ∧ ∅ ∈ On ∧ On = On))
61, 2, 4, 5mpbir3an 1179 1 Lim On
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  c0 3424   cuni 3811  Ord word 4364  Oncon0 4365  Lim wlim 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373
This theorem is referenced by: (None)
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