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Theorem ordelss 4142
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 4141 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 3892 . . 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 1434  wss 2974  Tr wtr 3883  Ord word 4125
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129
This theorem is referenced by:  ordelord  4144  onelss  4150  ordsuc  4314  smores3  5942  tfrlem1  5957  tfrlemisucaccv  5974  tfrlemiubacc  5979  tfr1onlemsucaccv  5990  tfr1onlemubacc  5995  tfrcllemsucaccv  6003  tfrcllemubacc  6008  nntri1  6140  nnsseleq  6145  fict  6403  infnfi  6429  ordiso2  6505  sizeinfuni  9801
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