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Theorem ordin 6193
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 6177 . . 3 (Ord 𝐴 → Tr 𝐴)
2 ordtr 6177 . . 3 (Ord 𝐵 → Tr 𝐵)
3 trin 5149 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
41, 2, 3syl2an 598 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴𝐵))
5 inss2 4159 . . 3 (𝐴𝐵) ⊆ 𝐵
6 trssord 6180 . . 3 ((Tr (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴𝐵))
75, 6mp3an2 1446 . 2 ((Tr (𝐴𝐵) ∧ Ord 𝐵) → Ord (𝐴𝐵))
84, 7sylancom 591 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  cin 3883  wss 3884  Tr wtr 5139  Ord word 6162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804  df-tr 5140  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166
This theorem is referenced by:  onin  6194  ordtri3or  6195  ordelinel  6261  smores  7976  smores2  7978  ordtypelem5  8974  ordtypelem7  8976
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