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Theorem ordelss 4589
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 4587 . 2  |-  ( Ord 
A  ->  Tr  A
)
2 trss 4303 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
32imp 419 . 2  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C_  A )
41, 3sylan 458 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3312   Tr wtr 4294   Ord word 4572
This theorem is referenced by:  onfr  4612  onelss  4615  ordtri2or2  4670  onfununi  6595  smores3  6607  tfrlem9a  6639  tz7.44-2  6657  tz7.44-3  6658  oaabslem  6878  oaabs2  6880  omabslem  6881  omabs  6882  findcard3  7342  nnsdomg  7358  ordiso2  7476  ordtypelem2  7480  ordtypelem6  7484  ordtypelem7  7485  cantnf  7641  cnfcomlem  7648  cardmin2  7877  infxpenlem  7887  iunfictbso  7987  dfac12lem2  8016  dfac12lem3  8017  unctb  8077  ackbij2lem1  8091  ackbij1lem3  8094  ackbij1lem18  8109  ackbij2  8115  ttukeylem6  8386  ttukeylem7  8387  alephexp1  8446  fpwwe2lem8  8504  pwfseqlem3  8527  pwcdandom  8534  fz1isolem  11702  onsuct0  26183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-ord 4576
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