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

Theorem ordin 4603
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 4587 . . 3  |-  ( Ord 
A  ->  Tr  A
)
2 ordtr 4587 . . 3  |-  ( Ord 
B  ->  Tr  B
)
3 trin 4304 . . 3  |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
41, 2, 3syl2an 464 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  Tr  ( A  i^i  B ) )
5 inss2 3554 . . 3  |-  ( A  i^i  B )  C_  B
6 trssord 4590 . . 3  |-  ( ( Tr  ( A  i^i  B )  /\  ( A  i^i  B )  C_  B  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
75, 6mp3an2 1267 . 2  |-  ( ( Tr  ( A  i^i  B )  /\  Ord  B
)  ->  Ord  ( A  i^i  B ) )
84, 7sylancom 649 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    i^i cin 3311    C_ wss 3312   Tr wtr 4294   Ord word 4572
This theorem is referenced by:  onin  4604  ordtri3or  4605  ordelinel  4671  smores  6605  smores2  6607  ordtypelem5  7480  ordtypelem7  7482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
  Copyright terms: Public domain W3C validator