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Theorem limelon 4411
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 4407 . . 3 (Lim 𝐴 → Ord 𝐴)
2 elong 4385 . . 3 (𝐴𝐵 → (𝐴 ∈ On ↔ Ord 𝐴))
31, 2imbitrrid 156 . 2 (𝐴𝐵 → (Lim 𝐴𝐴 ∈ On))
43imp 124 1 ((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2158  Ord word 4374  Oncon0 4375  Lim wlim 4376
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-in 3147  df-ss 3154  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-ilim 4381
This theorem is referenced by: (None)
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