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Theorem oneli 4680
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  e.  On
Assertion
Ref Expression
oneli  |-  ( B  e.  A  ->  B  e.  On )

Proof of Theorem oneli
StepHypRef Expression
1 on.1 . 2  |-  A  e.  On
2 onelon 4598 . 2  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
31, 2mpan 652 1  |-  ( B  e.  A  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   Oncon0 4573
This theorem is referenced by:  onssneli  4682  oawordeulem  6788  rankuni  7778  tcrank  7797  cardne  7841  cardval2  7867  alephsuc2  7950  cfsmolem  8139  cfcof  8143  alephreg  8446  pwcfsdom  8447  tskcard  8645  onsucconi  26135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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