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Theorem ordelord 4144
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 2142 . . . . 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 4135 . . . 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 4130 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
65simprbi 269 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
76r19.21bi 2450 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Tr  x )
8 ordelss 4142 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  C_  A )
9 simpl 107 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  A )
10 trssord 4143 . . . 4  |-  ( ( Tr  x  /\  x  C_  A  /\  Ord  A
)  ->  Ord  x )
117, 8, 9, 10syl3anc 1170 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
124, 11vtoclg 2659 . 2  |-  ( B  e.  A  ->  (
( Ord  A  /\  B  e.  A )  ->  Ord  B ) )
1312anabsi7 546 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349    C_ wss 2974   Tr wtr 3883   Ord word 4125
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129
This theorem is referenced by:  tron  4145  ordelon  4146  ordsucg  4254  ordwe  4326  smores  5941
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