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Theorem ordelss 4204
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 4203 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 3943 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 122 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 277 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  wss 2999  Tr wtr 3934  Ord word 4187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3652  df-tr 3935  df-iord 4191
This theorem is referenced by:  ordelord  4206  onelss  4212  ordsuc  4377  smores3  6050  tfrlem1  6065  tfrlemisucaccv  6082  tfrlemiubacc  6087  tfr1onlemsucaccv  6098  tfr1onlemubacc  6103  tfrcllemsucaccv  6111  tfrcllemubacc  6116  nntri1  6249  nnsseleq  6254  fict  6574  infnfi  6601  isinfinf  6603  ordiso2  6718  hashinfuni  10173
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