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Theorem oneli 4255
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 4211 . 2  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
31, 2mpan 415 1  |-  ( B  e.  A  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   Oncon0 4190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195
This theorem is referenced by:  nnon  4424
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