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Theorem onelssi 4358
Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onelssi  |-  ( B  e.  A  ->  B  C_  A )

Proof of Theorem onelssi
StepHypRef Expression
1 on.1 . 2  |-  A  e.  On
2 onelss 4316 . 2  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
31, 2ax-mp 5 1  |-  ( B  e.  A  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1481    C_ wss 3075   Oncon0 4292
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297
This theorem is referenced by:  onelini  4359  oneluni  4360  omp1eomlem  6986  enumctlemm  7006  ennnfonelemdc  11946  ennnfonelemg  11950  ctinfom  11975  2o01f  13362  isomninnlem  13398  iswomninnlem  13415  ismkvnnlem  13417
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