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Theorem onelss 4214
Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
onelss  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )

Proof of Theorem onelss
StepHypRef Expression
1 eloni 4202 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4206 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
32ex 113 . 2  |-  ( Ord 
A  ->  ( B  e.  A  ->  B  C_  A ) )
41, 3syl 14 1  |-  ( A  e.  On  ->  ( B  e.  A  ->  B 
C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438    C_ wss 2999   Ord word 4189   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-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:  onelssi  4256  ssorduni  4304  onsucelsucr  4325  tfisi  4402  tfrlem9  6084  nntri2or2  6259  phpelm  6580
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