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Theorem limelon 4328
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 4324 . . 3 (Lim 𝐴 → Ord 𝐴)
2 elong 4302 . . 3 (𝐴𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2syl5ibr 155 . 2 (𝐴𝐵 → (Lim 𝐴𝐴 ∈ On))
43imp 123 1 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1481  Ord word 4291  Oncon0 4292  Lim wlim 4293
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297  df-ilim 4298
This theorem is referenced by: (None)
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