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Theorem trssord 4376
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 4363 . . . . . . 7 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
21simprbi 275 . . . . . 6 (Ord 𝐵 → ∀𝑥𝐵 Tr 𝑥)
3 ssralv 3219 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵 Tr 𝑥 → ∀𝑥𝐴 Tr 𝑥))
42, 3syl5 32 . . . . 5 (𝐴𝐵 → (Ord 𝐵 → ∀𝑥𝐴 Tr 𝑥))
54imp 124 . . . 4 ((𝐴𝐵 ∧ Ord 𝐵) → ∀𝑥𝐴 Tr 𝑥)
65anim2i 342 . . 3 ((Tr 𝐴 ∧ (𝐴𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
763impb 1199 . 2 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
8 dford3 4363 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
97, 8sylibr 134 1 ((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wral 2455  wss 3129  Tr wtr 4098  Ord word 4358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3135  df-ss 3142  df-iord 4362
This theorem is referenced by:  ordelord  4377  ordin  4381  ssorduni  4482  ordtriexmidlem  4514  ordtri2or2exmidlem  4521  onsucelsucexmidlem  4524  ordsuc  4558
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