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Theorem ordin 5670
 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 5654 . . 3 (Ord 𝐴 → Tr 𝐴)
2 ordtr 5654 . . 3 (Ord 𝐵 → Tr 𝐵)
3 trin 4691 . . 3 ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
41, 2, 3syl2an 493 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴𝐵))
5 inss2 3796 . . 3 (𝐴𝐵) ⊆ 𝐵
6 trssord 5657 . . 3 ((Tr (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴𝐵))
75, 6mp3an2 1404 . 2 ((Tr (𝐴𝐵) ∧ Ord 𝐵) → Ord (𝐴𝐵))
84, 7sylancom 698 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∩ cin 3539   ⊆ wss 3540  Tr wtr 4680  Ord word 5639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643 This theorem is referenced by:  onin  5671  ordtri3or  5672  ordelinel  5742  ordelinelOLD  5743  smores  7336  smores2  7338  ordtypelem5  8310  ordtypelem7  8312
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