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Theorem onelssi 4412
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 4370 . 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 2141    C_ wss 3121   Oncon0 4346
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351
This theorem is referenced by:  onelini  4413  oneluni  4414  omp1eomlem  7067  enumctlemm  7087  ennnfonelemdc  12341  ennnfonelemg  12345  ctinfom  12370  2o01f  13989  isomninnlem  14022  iswomninnlem  14041  ismkvnnlem  14044
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