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Theorem oneli 4437
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 4354 . 2  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
31, 2mpan 654 1  |-  ( B  e.  A  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621   Oncon0 4329
This theorem is referenced by:  onssneli  4439  oawordeulem  6485  rankuni  7468  tcrank  7487  cardne  7531  cardval2  7557  alephsuc2  7640  cfsmolem  7829  cfcof  7833  alephreg  8137  pwcfsdom  8138  tskcard  8336  onsucconi  24216
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333
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