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Theorem oneli 4350
Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
oneli (𝐵𝐴𝐵 ∈ On)

Proof of Theorem oneli
StepHypRef Expression
1 on.1 . 2 𝐴 ∈ On
2 onelon 4306 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
31, 2mpan 420 1 (𝐵𝐴𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  Oncon0 4285
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290
This theorem is referenced by:  nnon  4523
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