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Theorem ordin 6347
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 6331 . . 3 (Ord 𝐴 → Tr 𝐴)
2 ordtr 6331 . . 3 (Ord 𝐵 → Tr 𝐵)
3 trin 5234 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
41, 2, 3syl2an 596 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴𝐵))
5 inss2 4189 . . 3 (𝐴𝐵) ⊆ 𝐵
6 trssord 6334 . . 3 ((Tr (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴𝐵))
75, 6mp3an2 1449 . 2 ((Tr (𝐴𝐵) ∧ Ord 𝐵) → Ord (𝐴𝐵))
84, 7sylancom 588 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  cin 3909  wss 3910  Tr wtr 5222  Ord word 6316
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-in 3917  df-ss 3927  df-uni 4866  df-tr 5223  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320
This theorem is referenced by:  onin  6348  ordtri3or  6349  ordelinel  6418  smores  8298  smores2  8300  ordtypelem5  9458  ordtypelem7  9460
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