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Theorem trssord 6331
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 5618 . . . . 5 (𝐴𝐵 → ( E We 𝐵 → E We 𝐴))
2 ordwe 6327 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 507 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 618 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1116 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 6317 . 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 397  w3a 1088  wss 3909  Tr wtr 5221   E cep 5534   We wwe 5585  Ord word 6313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-v 3446  df-in 3916  df-ss 3926  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-ord 6317
This theorem is referenced by:  ordin  6344  ssorduni  7704  ordsuci  7734  sucexeloniOLD  7736  suceloniOLD  7738  ordom  7803  ordtypelem2  9389  hartogs  9414  card2on  9424  tskwe  9820  ondomon  10433  dford3lem2  41253  dford3  41254  iunord  46912
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