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Theorem nlimsucg 4814
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 4632 . . . 4  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 4786 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 204 . . 3  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4633 . . 3  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
5 ordunisuc 4804 . . . . 5  |-  ( Ord 
A  ->  U. suc  A  =  A )
65eqeq2d 2446 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  <->  suc 
A  =  A ) )
7 ordirr 4591 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
8 eleq2 2496 . . . . . . 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 4651 . . . . . 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 1652    e. wcel 1725   U.cuni 4007   Ord word 4572   Lim wlim 4574   suc csuc 4575
This theorem is referenced by:  tz7.44-2  6657  rankxpsuc  7798  dfrdg2  25415  dfrdg4  25787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
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