MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordin Unicode version

Theorem ordin 4359
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 4343 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 ordtr 4343 . . 3  |-  ( Ord 
B  ->  Tr  B
)
3 trin 4063 . . 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 3332 . . 3  |-  ( A  i^i  B )  C_  B
6 trssord 4346 . . 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 3093    C_ wss 3094   Tr wtr 4053   Ord word 4328
This theorem is referenced by:  onin  4360  ordtri3or  4361  ordelinel  4428  smores  6302  smores2  6304  ordtypelem5  7170  ordtypelem7  7172
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-v 2742  df-in 3101  df-ss 3108  df-uni 3769  df-tr 4054  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332
  Copyright terms: Public domain W3C validator