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Theorem nlimsucg 6662
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 4937 . . . 4  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 6634 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 212 . . 3  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4938 . . 3  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
5 ordunisuc 6652 . . . . 5  |-  ( Ord 
A  ->  U. suc  A  =  A )
65eqeq2d 2481 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  <->  suc 
A  =  A ) )
7 ordirr 4896 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
8 eleq2 2540 . . . . . . 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 4956 . . . . . 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 1379    e. wcel 1767   U.cuni 4245   Ord word 4877   Lim wlim 4879   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884
This theorem is referenced by:  tz7.44-2  7074  rankxpsuc  8301  dfrdg2  29081  dfrdg4  29453
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