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Theorem ordelord 4399
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 2252 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
21anbi2d 464 . . . 4  |-  ( x  =  B  ->  (
( Ord  A  /\  x  e.  A )  <->  ( Ord  A  /\  B  e.  A ) ) )
3 ordeq 4390 . . . 4  |-  ( x  =  B  ->  ( Ord  x  <->  Ord  B ) )
42, 3imbi12d 234 . . 3  |-  ( x  =  B  ->  (
( ( Ord  A  /\  x  e.  A
)  ->  Ord  x )  <-> 
( ( Ord  A  /\  B  e.  A
)  ->  Ord  B ) ) )
5 dford3 4385 . . . . . 6  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
65simprbi 275 . . . . 5  |-  ( Ord 
A  ->  A. x  e.  A  Tr  x
)
76r19.21bi 2578 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Tr  x )
8 ordelss 4397 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  x  C_  A )
9 simpl 109 . . . 4  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  A )
10 trssord 4398 . . . 4  |-  ( ( Tr  x  /\  x  C_  A  /\  Ord  A
)  ->  Ord  x )
117, 8, 9, 10syl3anc 1249 . . 3  |-  ( ( Ord  A  /\  x  e.  A )  ->  Ord  x )
124, 11vtoclg 2812 . 2  |-  ( B  e.  A  ->  (
( Ord  A  /\  B  e.  A )  ->  Ord  B ) )
1312anabsi7 581 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468    C_ wss 3144   Tr wtr 4116   Ord word 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384
This theorem is referenced by:  tron  4400  ordelon  4401  ordsucg  4519  ordwe  4593  smores  6318
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