ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordelord Unicode version

Theorem ordelord 4208
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
Assertion
Ref Expression
ordelord  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )

Proof of Theorem ordelord
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2150 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21anbi2d 452 . . . 4  |-  ( x  =  B  ->  (
( Ord  A  /\  x  e.  A )  <->  ( Ord  A  /\  B  e.  A ) ) )
3 ordeq 4199 . . . 4  |-  ( x  =  B  ->  ( Ord  x  <->  Ord  B ) )
42, 3imbi12d 232 . . 3  |-  ( x  =  B  ->  (
( ( Ord  A  /\  x  e.  A
)  ->  Ord  x )  <-> 
( ( Ord  A  /\  B  e.  A
)  ->  Ord  B ) ) )
5 dford3 4194 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
65simprbi 269 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
76r19.21bi 2461 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Tr  x )
8 ordelss 4206 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  C_  A )
9 simpl 107 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  A )
10 trssord 4207 . . . 4  |-  ( ( Tr  x  /\  x  C_  A  /\  Ord  A
)  ->  Ord  x )
117, 8, 9, 10syl3anc 1174 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
124, 11vtoclg 2679 . 2  |-  ( B  e.  A  ->  (
( Ord  A  /\  B  e.  A )  ->  Ord  B ) )
1312anabsi7 548 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359    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-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193
This theorem is referenced by:  tron  4209  ordelon  4210  ordsucg  4319  ordwe  4391  smores  6057
  Copyright terms: Public domain W3C validator