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Theorem limon 4497
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 4470 . 2 Ord On
2 0elon 4377 . 2 ∅ ∈ On
3 unon 4495 . . 3 On = On
43eqcomi 2174 . 2 On = On
5 dflim2 4355 . 2 (Lim On ↔ (Ord On ∧ ∅ ∈ On ∧ On = On))
61, 2, 4, 5mpbir3an 1174 1 Lim On
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wcel 2141  c0 3414   cuni 3796  Ord word 4347  Oncon0 4348  Lim wlim 4349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356
This theorem is referenced by: (None)
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