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

Theorem ordin 4422
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 4406 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 ordtr 4406 . . 3  |-  ( Ord 
B  ->  Tr  B
)
3 trin 4123 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
41, 2, 3syl2an 463 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  Tr  ( A  i^i  B ) )
5 inss2 3390 . . 3  |-  ( A  i^i  B )  C_  B
6 trssord 4409 . . 3  |-  ( ( Tr  ( A  i^i  B )  /\  ( A  i^i  B )  C_  B  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
75, 6mp3an2 1265 . 2  |-  ( ( Tr  ( A  i^i  B )  /\  Ord  B
)  ->  Ord  ( A  i^i  B ) )
84, 7sylancom 648 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    i^i cin 3151    C_ wss 3152   Tr wtr 4113   Ord word 4391
This theorem is referenced by:  onin  4423  ordtri3or  4424  ordelinel  4491  smores  6369  smores2  6371  ordtypelem5  7237  ordtypelem7  7239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
  Copyright terms: Public domain W3C validator