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Theorem nlimsucg 4317
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 4158 . . . . . 6  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 4314 . . . . . 6  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 132 . . . . 5  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4159 . . . . 5  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
53, 4jca 300 . . . 4  |-  ( Lim 
suc  A  ->  ( Ord 
A  /\  suc  A  = 
U. suc  A )
)
6 ordtr 4141 . . . . . . . 8  |-  ( Ord 
A  ->  Tr  A
)
7 unisucg 4177 . . . . . . . . 9  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
87biimpa 290 . . . . . . . 8  |-  ( ( A  e.  V  /\  Tr  A )  ->  U. suc  A  =  A )
96, 8sylan2 280 . . . . . . 7  |-  ( ( A  e.  V  /\  Ord  A )  ->  U. suc  A  =  A )
109eqeq2d 2093 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  <->  suc  A  =  A ) )
11 ordirr 4293 . . . . . . . . 9  |-  ( Ord 
A  ->  -.  A  e.  A )
12 eleq2 2143 . . . . . . . . . 10  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
1312notbid 625 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
1411, 13syl5ibrcom 155 . . . . . . . 8  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
15 sucidg 4179 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  suc  A )
1615con3i 595 . . . . . . . 8  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1714, 16syl6 33 . . . . . . 7  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
1817adantl 271 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  A  ->  -.  A  e.  V
) )
1910, 18sylbid 148 . . . . 5  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V ) )
2019expimpd 355 . . . 4  |-  ( A  e.  V  ->  (
( Ord  A  /\  suc  A  =  U. suc  A )  ->  -.  A  e.  V ) )
215, 20syl5 32 . . 3  |-  ( A  e.  V  ->  ( Lim  suc  A  ->  -.  A  e.  V )
)
2221con2d 587 . 2  |-  ( A  e.  V  ->  ( A  e.  V  ->  -. 
Lim  suc  A ) )
2322pm2.43i 48 1  |-  ( A  e.  V  ->  -.  Lim  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   U.cuni 3609   Tr wtr 3883   Ord word 4125   Lim wlim 4127   suc csuc 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610  df-tr 3884  df-iord 4129  df-ilim 4132  df-suc 4134
This theorem is referenced by: (None)
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