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Theorem trssord 4145
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 4132 . . . . . . 7 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
21simprbi 264 . . . . . 6 (Ord 𝐵 → ∀𝑥𝐵 Tr 𝑥)
3 ssralv 3032 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥))
42, 3syl5 32 . . . . 5 (𝐴𝐵 → (Ord 𝐵 → ∀𝑥𝐴 Tr 𝑥))
54imp 119 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → ∀𝑥𝐴 Tr 𝑥)
65anim2i 328 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
763impb 1111 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
8 dford3 4132 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
97, 8sylibr 141 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wral 2323  wss 2945  Tr wtr 3882  Ord word 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-in 2952  df-ss 2959  df-iord 4131
This theorem is referenced by:  ordelord  4146  ordin  4150  ssorduni  4241  ordtriexmidlem  4273  ordtri2or2exmidlem  4279  onsucelsucexmidlem  4282  ordsuc  4315
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