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Theorem ordelss 6337
Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
Assertion
Ref Expression
ordelss ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)

Proof of Theorem ordelss
StepHypRef Expression
1 ordtr 6335 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 5237 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 408 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 581 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wss 3914  Tr wtr 5226  Ord word 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-uni 4870  df-tr 5227  df-ord 6324
This theorem is referenced by:  onfr  6360  onelss  6363  ordtri2or2  6420  onfununi  8291  smores3  8303  tfrlem1  8326  tfrlem9a  8336  tz7.44-2  8357  tz7.44-3  8358  oaabslem  8597  oaabs2  8599  omabslem  8600  omabs  8601  findcard3  9235  findcard3OLD  9236  nnsdomg  9252  nnsdomgOLD  9253  ordiso2  9459  ordtypelem2  9463  ordtypelem6  9467  ordtypelem7  9468  cantnf  9637  cnfcomlem  9643  ttrcltr  9660  cardmin2  9943  infxpenlem  9957  iunfictbso  10058  dfac12lem2  10088  dfac12lem3  10089  unctb  10149  ackbij2lem1  10163  ackbij1lem3  10166  ackbij1lem18  10181  ackbij2  10187  ttukeylem6  10458  ttukeylem7  10459  alephexp1  10523  fpwwe2lem7  10581  pwfseqlem3  10604  pwdjundom  10611  fz1isolem  14369  noinfbday  27091  onsuct0  34966  finxpreclem4  35915  nadd2rabtr  41747  grur1cld  42604
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