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Theorem nlimsucg 4565
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 4395 . . . . . 6  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 4562 . . . . . 6  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 134 . . . . 5  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 4396 . . . . 5  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
53, 4jca 306 . . . 4  |-  ( Lim 
suc  A  ->  ( Ord 
A  /\  suc  A  = 
U. suc  A )
)
6 ordtr 4378 . . . . . . . 8  |-  ( Ord 
A  ->  Tr  A
)
7 unisucg 4414 . . . . . . . . 9  |-  ( A  e.  V  ->  ( Tr  A  <->  U. suc  A  =  A ) )
87biimpa 296 . . . . . . . 8  |-  ( ( A  e.  V  /\  Tr  A )  ->  U. suc  A  =  A )
96, 8sylan2 286 . . . . . . 7  |-  ( ( A  e.  V  /\  Ord  A )  ->  U. suc  A  =  A )
109eqeq2d 2189 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  <->  suc  A  =  A ) )
11 ordirr 4541 . . . . . . . . 9  |-  ( Ord 
A  ->  -.  A  e.  A )
12 eleq2 2241 . . . . . . . . . 10  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
1312notbid 667 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
1411, 13syl5ibrcom 157 . . . . . . . 8  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
15 sucidg 4416 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  suc  A )
1615con3i 632 . . . . . . . 8  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1714, 16syl6 33 . . . . . . 7  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
1817adantl 277 . . . . . 6  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  A  ->  -.  A  e.  V
) )
1910, 18sylbid 150 . . . . 5  |-  ( ( A  e.  V  /\  Ord  A )  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V ) )
2019expimpd 363 . . . 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 624 . 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 104    = wceq 1353    e. wcel 2148   U.cuni 3809   Tr wtr 4101   Ord word 4362   Lim wlim 4364   suc csuc 4365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-ilim 4369  df-suc 4371
This theorem is referenced by: (None)
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