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Theorem ordelss 4271
Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
Assertion
Ref Expression
ordelss  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )

Proof of Theorem ordelss
StepHypRef Expression
1 ordtr 4270 . 2  |-  ( Ord 
A  ->  Tr  A
)
2 trss 4005 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
32imp 123 . 2  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C_  A )
41, 3sylan 281 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465    C_ wss 3041   Tr wtr 3996   Ord word 4254
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-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258
This theorem is referenced by:  ordelord  4273  onelss  4279  ordsuc  4448  smores3  6158  tfrlem1  6173  tfrlemisucaccv  6190  tfrlemiubacc  6195  tfr1onlemsucaccv  6206  tfr1onlemubacc  6211  tfrcllemsucaccv  6219  tfrcllemubacc  6224  nntri1  6360  nnsseleq  6365  fict  6730  infnfi  6757  isinfinf  6759  ordiso2  6888  hashinfuni  10491
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