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Theorem ordin 4212
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 4205 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 ordtr 4205 . . 3  |-  ( Ord 
B  ->  Tr  B
)
3 trin 3946 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
41, 2, 3syl2an 283 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  Tr  ( A  i^i  B ) )
5 inss2 3221 . . 3  |-  ( A  i^i  B )  C_  B
6 trssord 4207 . . 3  |-  ( ( Tr  ( A  i^i  B )  /\  ( A  i^i  B )  C_  B  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
75, 6mp3an2 1261 . 2  |-  ( ( Tr  ( A  i^i  B )  /\  Ord  B
)  ->  Ord  ( A  i^i  B ) )
84, 7sylancom 411 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    i^i cin 2998    C_ wss 2999   Tr wtr 3936   Ord word 4189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193
This theorem is referenced by:  onin  4213  smores  6057  smores2  6059
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