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Theorem limsuc2 30579
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 6650 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
21biimpa 484 . . . 4  |-  ( ( Ord  A  /\  A  =  U. A )  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 4936 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2529 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspccva 3206 . . . 4  |-  ( ( A. x  e.  A  suc  x  e.  A  /\  B  e.  A )  ->  suc  B  e.  A
)
62, 5sylan 471 . . 3  |-  ( ( ( Ord  A  /\  A  =  U. A )  /\  B  e.  A
)  ->  suc  B  e.  A )
76ex 434 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  ->  suc  B  e.  A
) )
8 ordtr 4885 . . . 4  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4955 . . . . 5  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 434 . . . 4  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
118, 10syl 16 . . 3  |-  ( Ord 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
1211adantr 465 . 2  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( suc  B  e.  A  ->  B  e.  A
) )
137, 12impbid 191 1  |-  ( ( Ord  A  /\  A  =  U. A )  -> 
( B  e.  A  <->  suc 
B  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   U.cuni 4238   Tr wtr 4533   Ord word 4870   suc csuc 4873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877
This theorem is referenced by:  aomclem4  30596
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