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Theorem ordelss 4116
 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 4115 . 2 (Ord 𝐴 → Tr 𝐴)
2 trss 3863 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
32imp 115 . 2 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
41, 3sylan 267 1 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393   ⊆ wss 2917  Tr wtr 3854  Ord word 4099 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103 This theorem is referenced by:  ordelord  4118  onelss  4124  ordsuc  4287  smores3  5908  tfrlem1  5923  tfrlemisucaccv  5939  tfrlemiubacc  5944  nntri1  6074  nnsseleq  6079  ordiso2  6357
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