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Theorem trssord 4272
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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dford3 4259 . . . . . . 7 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
21simprbi 273 . . . . . 6 (Ord 𝐵 → ∀𝑥𝐵 Tr 𝑥)
3 ssralv 3131 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥))
42, 3syl5 32 . . . . 5 (𝐴𝐵 → (Ord 𝐵 → ∀𝑥𝐴 Tr 𝑥))
54imp 123 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → ∀𝑥𝐴 Tr 𝑥)
65anim2i 339 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
763impb 1162 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
8 dford3 4259 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
97, 8sylibr 133 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wral 2393  wss 3041  Tr wtr 3996  Ord word 4254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-in 3047  df-ss 3054  df-iord 4258
This theorem is referenced by:  ordelord  4273  ordin  4277  ssorduni  4373  ordtriexmidlem  4405  ordtri2or2exmidlem  4411  onsucelsucexmidlem  4414  ordsuc  4448
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