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Theorem nlimsucg 6453
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 4778 . . . 4  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 6425 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 212 . . 3  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4779 . . 3  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
5 ordunisuc 6443 . . . . 5  |-  ( Ord 
A  ->  U. suc  A  =  A )
65eqeq2d 2454 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  <->  suc 
A  =  A ) )
7 ordirr 4737 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
8 eleq2 2504 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
98notbid 294 . . . . . 6  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
107, 9syl5ibrcom 222 . . . . 5  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
11 sucidg 4797 . . . . . 6  |-  ( A  e.  V  ->  A  e.  suc  A )
1211con3i 135 . . . . 5  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1310, 12syl6 33 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
146, 13sylbid 215 . . 3  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V
) )
153, 4, 14sylc 60 . 2  |-  ( Lim 
suc  A  ->  -.  A  e.  V )
1615con2i 120 1  |-  ( A  e.  V  ->  -.  Lim  suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   U.cuni 4091   Ord word 4718   Lim wlim 4720   suc csuc 4721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725
This theorem is referenced by:  tz7.44-2  6863  rankxpsuc  8089  dfrdg2  27609  dfrdg4  27981
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