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Theorem ordelord 4303
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 2202 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21anbi2d 459 . . . 4  |-  ( x  =  B  ->  (
( Ord  A  /\  x  e.  A )  <->  ( Ord  A  /\  B  e.  A ) ) )
3 ordeq 4294 . . . 4  |-  ( x  =  B  ->  ( Ord  x  <->  Ord  B ) )
42, 3imbi12d 233 . . 3  |-  ( x  =  B  ->  (
( ( Ord  A  /\  x  e.  A
)  ->  Ord  x )  <-> 
( ( Ord  A  /\  B  e.  A
)  ->  Ord  B ) ) )
5 dford3 4289 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
65simprbi 273 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
76r19.21bi 2520 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Tr  x )
8 ordelss 4301 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  C_  A )
9 simpl 108 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  A )
10 trssord 4302 . . . 4  |-  ( ( Tr  x  /\  x  C_  A  /\  Ord  A
)  ->  Ord  x )
117, 8, 9, 10syl3anc 1216 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
124, 11vtoclg 2746 . 2  |-  ( B  e.  A  ->  (
( Ord  A  /\  B  e.  A )  ->  Ord  B ) )
1312anabsi7 570 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416    C_ wss 3071   Tr wtr 4026   Ord word 4284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288
This theorem is referenced by:  tron  4304  ordelon  4305  ordsucg  4418  ordwe  4490  smores  6189
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