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Theorem ordelord 4364
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 2233 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21anbi2d 461 . . . 4  |-  ( x  =  B  ->  (
( Ord  A  /\  x  e.  A )  <->  ( Ord  A  /\  B  e.  A ) ) )
3 ordeq 4355 . . . 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 4350 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
65simprbi 273 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
76r19.21bi 2558 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Tr  x )
8 ordelss 4362 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  C_  A )
9 simpl 108 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  A )
10 trssord 4363 . . . 4  |-  ( ( Tr  x  /\  x  C_  A  /\  Ord  A
)  ->  Ord  x )
117, 8, 9, 10syl3anc 1233 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
124, 11vtoclg 2790 . 2  |-  ( B  e.  A  ->  (
( Ord  A  /\  B  e.  A )  ->  Ord  B ) )
1312anabsi7 576 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   A.wral 2448    C_ wss 3121   Tr wtr 4085   Ord word 4345
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349
This theorem is referenced by:  tron  4365  ordelon  4366  ordsucg  4484  ordwe  4558  smores  6269
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