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Theorem trssord 6395
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 6391 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 504 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 615 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1112 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 6381 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
75, 6sylibr 233 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wss 3947  Tr wtr 5272   E cep 5587   We wwe 5638  Ord word 6377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-ral 3052  df-ss 3964  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-ord 6381
This theorem is referenced by:  ordin  6408  ssorduni  7789  ordsuci  7819  sucexeloniOLD  7821  suceloniOLD  7823  ordom  7888  ordtypelem2  9564  hartogs  9589  card2on  9599  tskwe  9995  ondomon  10608  dford3lem2  42703  dford3  42704  iunord  48440
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