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Theorem ordelss 4408
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 4406 . 2  |-  ( Ord 
A  ->  Tr  A
)
2 trss 4122 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
32imp 418 . 2  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C_  A )
41, 3sylan 457 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    C_ wss 3152   Tr wtr 4113   Ord word 4391
This theorem is referenced by:  onfr  4431  onelss  4434  ordtri2or2  4489  onfununi  6358  smores3  6370  tfrlem9a  6402  tz7.44-2  6420  tz7.44-3  6421  oaabslem  6641  oaabs2  6643  omabslem  6644  omabs  6645  findcard3  7100  nnsdomg  7116  ordiso2  7230  ordtypelem2  7234  ordtypelem6  7238  ordtypelem7  7239  cantnf  7395  cnfcomlem  7402  cardmin2  7631  infxpenlem  7641  iunfictbso  7741  dfac12lem2  7770  dfac12lem3  7771  unctb  7831  ackbij2lem1  7845  ackbij1lem3  7848  ackbij1lem18  7863  ackbij2  7869  ttukeylem6  8141  ttukeylem7  8142  alephexp1  8201  fpwwe2lem8  8259  pwfseqlem3  8282  pwcdandom  8289  fz1isolem  11399  onsuct0  24880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-ord 4395
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