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Theorem trssord 5958
Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
Assertion
Ref Expression
trssord ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Proof of Theorem trssord
StepHypRef Expression
1 wess 5299 . . . . 5 (𝐴𝐵 → ( E We 𝐵 → E We 𝐴))
2 ordwe 5954 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 502 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 611 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1144 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 5944 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
75, 6sylibr 226 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108  wss 3769  Tr wtr 4945   E cep 5224   We wwe 5270  Ord word 5940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-in 3776  df-ss 3783  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944
This theorem is referenced by:  ordin  5971  ssorduni  7219  suceloni  7247  ordom  7308  ordtypelem2  8666  hartogs  8691  card2on  8701  tskwe  9062  ondomon  9673  dford3lem2  38379  dford3  38380  iunord  43221
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