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Theorem oneli 4391
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 4310 . 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 4285
This theorem is referenced by:  onssneli  4393  oawordeulem  6438  rankuni  7419  tcrank  7438  cardne  7482  cardval2  7508  alephsuc2  7591  cfsmolem  7780  cfcof  7784  alephreg  8084  pwcfsdom  8085  tskcard  8283  onsucconi  24050
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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