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Theorem trssord 6401
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 5671 . . . . 5 (𝐴𝐵 → ( E We 𝐵 → E We 𝐴))
2 ordwe 6397 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 505 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 617 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1115 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 6387 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
75, 6sylibr 234 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wss 3951  Tr wtr 5259   E cep 5583   We wwe 5636  Ord word 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ral 3062  df-ss 3968  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by:  ordin  6414  ssorduni  7799  ordsuci  7828  sucexeloniOLD  7830  suceloniOLD  7832  ordom  7897  ordtypelem2  9559  hartogs  9584  card2on  9594  tskwe  9990  ondomon  10603  dford3lem2  43039  dford3  43040  iunord  49195
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