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Theorem ordin 4142
Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
ordin ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))

Proof of Theorem ordin
StepHypRef Expression
1 ordtr 4135 . . 3 (Ord 𝐴 → Tr 𝐴)
2 ordtr 4135 . . 3 (Ord 𝐵 → Tr 𝐵)
3 trin 3887 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
41, 2, 3syl2an 283 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴𝐵))
5 inss2 3188 . . 3 (𝐴𝐵) ⊆ 𝐵
6 trssord 4137 . . 3 ((Tr (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴𝐵))
75, 6mp3an2 1257 . 2 ((Tr (𝐴𝐵) ∧ Ord 𝐵) → Ord (𝐴𝐵))
84, 7sylancom 411 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  cin 2973  wss 2974  Tr wtr 3877  Ord word 4119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878  df-iord 4123
This theorem is referenced by:  onin  4143  smores  5935  smores2  5937
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