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Theorem nlimsucg 4764
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg  |-  ( A  e.  V  ->  -.  Lim  suc  A )

Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 4583 . . . 4  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 4736 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 204 . . 3  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4584 . . 3  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
5 ordunisuc 4754 . . . . 5  |-  ( Ord 
A  ->  U. suc  A  =  A )
65eqeq2d 2400 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  <->  suc 
A  =  A ) )
7 ordirr 4542 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
8 eleq2 2450 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
98notbid 286 . . . . . 6  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
107, 9syl5ibrcom 214 . . . . 5  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
11 sucidg 4602 . . . . . 6  |-  ( A  e.  V  ->  A  e.  suc  A )
1211con3i 129 . . . . 5  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1310, 12syl6 31 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
146, 13sylbid 207 . . 3  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V
) )
153, 4, 14sylc 58 . 2  |-  ( Lim 
suc  A  ->  -.  A  e.  V )
1615con2i 114 1  |-  ( A  e.  V  ->  -.  Lim  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   U.cuni 3959   Ord word 4523   Lim wlim 4525   suc csuc 4526
This theorem is referenced by:  tz7.44-2  6603  rankxpsuc  7741  dfrdg2  25178  dfrdg4  25515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-tr 4246  df-eprel 4437  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530
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