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Theorem ordin 4421
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  ( A  i^i  B ) )

Proof of Theorem ordin
StepHypRef Expression
1 ordtr 4405 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 ordtr 4405 . . 3  |-  ( Ord 
B  ->  Tr  B
)
3 trin 4124 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
41, 2, 3syl2an 465 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  Tr  ( A  i^i  B ) )
5 inss2 3391 . . 3  |-  ( A  i^i  B )  C_  B
6 trssord 4408 . . 3  |-  ( ( Tr  ( A  i^i  B )  /\  ( A  i^i  B )  C_  B  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
75, 6mp3an2 1270 . 2  |-  ( ( Tr  ( A  i^i  B )  /\  Ord  B
)  ->  Ord  ( A  i^i  B ) )
84, 7sylancom 651 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    i^i cin 3152    C_ wss 3153   Tr wtr 4114   Ord word 4390
This theorem is referenced by:  onin  4422  ordtri3or  4423  ordelinel  4490  smores  6364  smores2  6366  ordtypelem5  7232  ordtypelem7  7234
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-v 2791  df-in 3160  df-ss 3167  df-uni 3829  df-tr 4115  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394
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