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Theorem limsuc2 27137
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 4635 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
21biimpa 470 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 4457 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2349 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspccva 2883 . . . 4  |-  ( ( A. x  e.  A  suc  x  e.  A  /\  B  e.  A )  ->  suc  B  e.  A
)
62, 5sylan 457 . . 3  |-  ( ( ( Ord  A  /\  A  =  U. A )  /\  B  e.  A
)  ->  suc  B  e.  A )
76ex 423 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  ->  suc  B  e.  A
) )
8 ordtr 4406 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4476 . . . . 5  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 423 . . . 4  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
118, 10syl 15 . . 3  |-  ( Ord 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
1211adantr 451 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( suc  B  e.  A  ->  B  e.  A
) )
137, 12impbid 183 1  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   U.cuni 3827   Tr wtr 4113   Ord word 4391   suc csuc 4394
This theorem is referenced by:  aomclem4  27154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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