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Theorem limsuc2 35899
Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Assertion
Ref Expression
limsuc2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )

Proof of Theorem limsuc2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ordunisuc2 6671 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
21biimpa 487 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 5488 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2513 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspccva 3149 . . . 4  |-  ( ( A. x  e.  A  suc  x  e.  A  /\  B  e.  A )  ->  suc  B  e.  A
)
62, 5sylan 474 . . 3  |-  ( ( ( Ord  A  /\  A  =  U. A )  /\  B  e.  A
)  ->  suc  B  e.  A )
76ex 436 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  ->  suc  B  e.  A
) )
8 ordtr 5437 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 5507 . . . . 5  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 436 . . . 4  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
118, 10syl 17 . . 3  |-  ( Ord 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
1211adantr 467 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( suc  B  e.  A  ->  B  e.  A
) )
137, 12impbid 194 1  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   U.cuni 4198   Tr wtr 4497   Ord word 5422   suc csuc 5425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427  df-suc 5429
This theorem is referenced by:  aomclem4  35915
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