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Theorem ordin 5914
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 5898 . . 3 (Ord 𝐴 → Tr 𝐴)
2 ordtr 5898 . . 3 (Ord 𝐵 → Tr 𝐵)
3 trin 4915 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
41, 2, 3syl2an 495 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴𝐵))
5 inss2 3977 . . 3 (𝐴𝐵) ⊆ 𝐵
6 trssord 5901 . . 3 ((Tr (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴𝐵))
75, 6mp3an2 1561 . 2 ((Tr (𝐴𝐵) ∧ Ord 𝐵) → Ord (𝐴𝐵))
84, 7sylancom 704 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  cin 3714  wss 3715  Tr wtr 4904  Ord word 5883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887
This theorem is referenced by:  onin  5915  ordtri3or  5916  ordelinel  5986  ordelinelOLD  5987  smores  7619  smores2  7621  ordtypelem5  8594  ordtypelem7  8596
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