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Theorem ordelss 6200
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 6198 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 5172 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 407 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 580 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wss 3933  Tr wtr 5163  Ord word 6183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-in 3940  df-ss 3949  df-uni 4831  df-tr 5164  df-ord 6187
This theorem is referenced by:  onfr  6223  onelss  6226  ordtri2or2  6280  onfununi  7967  smores3  7979  tfrlem1  8001  tfrlem9a  8011  tz7.44-2  8032  tz7.44-3  8033  oaabslem  8259  oaabs2  8261  omabslem  8262  omabs  8263  findcard3  8749  nnsdomg  8765  ordiso2  8967  ordtypelem2  8971  ordtypelem6  8975  ordtypelem7  8976  cantnf  9144  cnfcomlem  9150  cardmin2  9415  infxpenlem  9427  iunfictbso  9528  dfac12lem2  9558  dfac12lem3  9559  unctb  9615  ackbij2lem1  9629  ackbij1lem3  9632  ackbij1lem18  9647  ackbij2  9653  ttukeylem6  9924  ttukeylem7  9925  alephexp1  9989  fpwwe2lem8  10047  pwfseqlem3  10070  pwdjundom  10077  fz1isolem  13807  onsuct0  33686  finxpreclem4  34557  grur1cld  40445
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