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Theorem limelon 4182
 Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
limelon ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)

Proof of Theorem limelon
StepHypRef Expression
1 limord 4178 . . 3 (Lim 𝐴 → Ord 𝐴)
2 elong 4156 . . 3 (𝐴𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2syl5ibr 154 . 2 (𝐴𝐵 → (Lim 𝐴𝐴 ∈ On))
43imp 122 1 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∈ wcel 1434  Ord word 4145  Oncon0 4146  Lim wlim 4147 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151  df-ilim 4152 This theorem is referenced by: (None)
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