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Theorem nlimsucg 4451
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 4287 . . . . . 6  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 4448 . . . . . 6  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 133 . . . . 5  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4288 . . . . 5  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
53, 4jca 304 . . . 4  |-  ( Lim 
suc  A  ->  ( Ord 
A  /\  suc  A  = 
U. suc  A )
)
6 ordtr 4270 . . . . . . . 8  |-  ( Ord 
A  ->  Tr  A
)
7 unisucg 4306 . . . . . . . . 9  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
87biimpa 294 . . . . . . . 8  |-  ( ( A  e.  V  /\  Tr  A )  ->  U. suc  A  =  A )
96, 8sylan2 284 . . . . . . 7  |-  ( ( A  e.  V  /\  Ord  A )  ->  U. suc  A  =  A )
109eqeq2d 2129 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  <->  suc  A  =  A ) )
11 ordirr 4427 . . . . . . . . 9  |-  ( Ord 
A  ->  -.  A  e.  A )
12 eleq2 2181 . . . . . . . . . 10  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
1312notbid 641 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
1411, 13syl5ibrcom 156 . . . . . . . 8  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
15 sucidg 4308 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  suc  A )
1615con3i 606 . . . . . . . 8  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1714, 16syl6 33 . . . . . . 7  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
1817adantl 275 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  A  ->  -.  A  e.  V
) )
1910, 18sylbid 149 . . . . 5  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V ) )
2019expimpd 360 . . . 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 598 . 2  |-  ( A  e.  V  ->  ( A  e.  V  ->  -. 
Lim  suc  A ) )
2322pm2.43i 49 1  |-  ( A  e.  V  ->  -.  Lim  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   U.cuni 3706   Tr wtr 3996   Ord word 4254   Lim wlim 4256   suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-ilim 4261  df-suc 4263
This theorem is referenced by: (None)
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