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Theorem limelon 4102
Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
limelon ((A B Lim A) → A On)

Proof of Theorem limelon
StepHypRef Expression
1 limord 4098 . . 3 (Lim A → Ord A)
2 elong 4076 . . 3 (A B → (A On ↔ Ord A))
31, 2syl5ibr 145 . 2 (A B → (Lim AA On))
43imp 115 1 ((A B Lim A) → A On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Ord word 4065  Oncon0 4066  Lim wlim 4067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-ilim 4072
This theorem is referenced by: (None)
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