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Theorem trssord 5699
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 5061 . . . . 5 (𝐴𝐵 → ( E We 𝐵 → E We 𝐴))
2 ordwe 5695 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 485 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 592 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1257 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 5685 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
75, 6sylibr 224 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wss 3555  Tr wtr 4712   E cep 4983   We wwe 5032  Ord word 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3562  df-ss 3569  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685
This theorem is referenced by:  ordin  5712  ssorduni  6932  suceloni  6960  ordom  7021  ordtypelem2  8368  hartogs  8393  card2on  8403  tskwe  8720  ondomon  9329  dford3lem2  37074  dford3  37075  iunord  41714
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