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Theorem ordelss 4424
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 4422 . 2  |-  ( Ord 
A  ->  Tr  A
)
2 trss 4138 . . 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 1696    C_ wss 3165   Tr wtr 4129   Ord word 4407
This theorem is referenced by:  onfr  4447  onelss  4450  ordtri2or2  4505  onfununi  6374  smores3  6386  tfrlem9a  6418  tz7.44-2  6436  tz7.44-3  6437  oaabslem  6657  oaabs2  6659  omabslem  6660  omabs  6661  findcard3  7116  nnsdomg  7132  ordiso2  7246  ordtypelem2  7250  ordtypelem6  7254  ordtypelem7  7255  cantnf  7411  cnfcomlem  7418  cardmin2  7647  infxpenlem  7657  iunfictbso  7757  dfac12lem2  7786  dfac12lem3  7787  unctb  7847  ackbij2lem1  7861  ackbij1lem3  7864  ackbij1lem18  7879  ackbij2  7885  ttukeylem6  8157  ttukeylem7  8158  alephexp1  8217  fpwwe2lem8  8275  pwfseqlem3  8298  pwcdandom  8305  fz1isolem  11415  onsuct0  24952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-ord 4411
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