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Theorem onelss 4279
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 4267 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelss 4271 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
32ex 114 . 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 1465    C_ wss 3041   Ord word 4254   Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  onelssi  4321  ssorduni  4373  onsucelsucr  4394  tfisi  4471  tfrlem9  6184  nntri2or2  6362  phpelm  6728  ennnfonelemk  11840
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