MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlimsucg Structured version   Visualization version   Unicode version

Theorem nlimsucg 6688
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 5489 . . . 4  |-  ( Lim 
suc  A  ->  Ord  suc  A )
2 ordsuc 6660 . . . 4  |-  ( Ord 
A  <->  Ord  suc  A )
31, 2sylibr 217 . . 3  |-  ( Lim 
suc  A  ->  Ord  A
)
4 limuni 5490 . . 3  |-  ( Lim 
suc  A  ->  suc  A  =  U. suc  A )
5 ordunisuc 6678 . . . . 5  |-  ( Ord 
A  ->  U. suc  A  =  A )
65eqeq2d 2481 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  <->  suc 
A  =  A ) )
7 ordirr 5448 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
8 eleq2 2538 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( A  e.  suc  A  <-> 
A  e.  A ) )
98notbid 301 . . . . . 6  |-  ( suc 
A  =  A  -> 
( -.  A  e. 
suc  A  <->  -.  A  e.  A ) )
107, 9syl5ibrcom 230 . . . . 5  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  suc  A ) )
11 sucidg 5508 . . . . . 6  |-  ( A  e.  V  ->  A  e.  suc  A )
1211con3i 142 . . . . 5  |-  ( -.  A  e.  suc  A  ->  -.  A  e.  V
)
1310, 12syl6 33 . . . 4  |-  ( Ord 
A  ->  ( suc  A  =  A  ->  -.  A  e.  V )
)
146, 13sylbid 223 . . 3  |-  ( Ord 
A  ->  ( suc  A  =  U. suc  A  ->  -.  A  e.  V
) )
153, 4, 14sylc 61 . 2  |-  ( Lim 
suc  A  ->  -.  A  e.  V )
1615con2i 124 1  |-  ( A  e.  V  ->  -.  Lim  suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    e. wcel 1904   U.cuni 4190   Ord word 5429   Lim wlim 5431   suc csuc 5432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436
This theorem is referenced by:  tz7.44-2  7143  rankxpsuc  8371  dfrdg2  30513  dfrdg4  30789
  Copyright terms: Public domain W3C validator