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Theorem trssord 5901
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 5253 . . . . 5 (𝐴𝐵 → ( E We 𝐵 → E We 𝐴))
2 ordwe 5897 . . . . 5 (Ord 𝐵 → E We 𝐵)
31, 2impel 486 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → E We 𝐴)
43anim2i 594 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
543impb 1108 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6 df-ord 5887 . 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 383  w3a 1072  wss 3715  Tr wtr 4904   E cep 5178   We wwe 5224  Ord word 5883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-in 3722  df-ss 3729  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887
This theorem is referenced by:  ordin  5914  ssorduni  7151  suceloni  7179  ordom  7240  ordtypelem2  8591  hartogs  8616  card2on  8626  tskwe  8986  ondomon  9597  dford3lem2  38114  dford3  38115  iunord  42950
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