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Theorem ordelss 6402
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 6400 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 5276 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 406 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 580 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wss 3963  Tr wtr 5265  Ord word 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-tr 5266  df-ord 6389
This theorem is referenced by:  onfr  6425  onelss  6428  ordtri2or2  6485  onfununi  8380  smores3  8392  tfrlem1  8415  tfrlem9a  8425  tz7.44-2  8446  tz7.44-3  8447  oaabslem  8684  oaabs2  8686  omabslem  8687  omabs  8688  findcard3  9316  findcard3OLD  9317  nnsdomg  9333  nnsdomgOLD  9334  ordiso2  9553  ordtypelem2  9557  ordtypelem6  9561  ordtypelem7  9562  cantnf  9731  cnfcomlem  9737  ttrcltr  9754  cardmin2  10037  infxpenlem  10051  iunfictbso  10152  dfac12lem2  10183  dfac12lem3  10184  unctb  10242  ackbij2lem1  10256  ackbij1lem3  10259  ackbij1lem18  10274  ackbij2  10280  ttukeylem6  10552  ttukeylem7  10553  alephexp1  10617  fpwwe2lem7  10675  pwfseqlem3  10698  pwdjundom  10705  fz1isolem  14497  noinfbday  27780  onsuct0  36424  finxpreclem4  37377  nadd2rabtr  43374  grur1cld  44228
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