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Theorem limsucncmp 24957
Description: The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Assertion
Ref Expression
limsucncmp  |-  ( Lim 
A  ->  -.  suc  A  e.  Comp )

Proof of Theorem limsucncmp
StepHypRef Expression
1 suceq 4473 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
21eleq1d 2362 . . 3  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( suc 
A  e.  Comp  <->  suc  if ( Lim  A ,  A ,  On )  e.  Comp ) )
32notbid 285 . 2  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( -. 
suc  A  e.  Comp  <->  -.  suc  if ( Lim  A ,  A ,  On )  e.  Comp ) )
4 limeq 4420 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
5 limeq 4420 . . . 4  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
6 limon 4643 . . . 4  |-  Lim  On
74, 5, 6elimhyp 3626 . . 3  |-  Lim  if ( Lim  A ,  A ,  On )
87limsucncmpi 24956 . 2  |-  -.  suc  if ( Lim  A ,  A ,  On )  e.  Comp
93, 8dedth 3619 1  |-  ( Lim 
A  ->  -.  suc  A  e.  Comp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   ifcif 3578   Oncon0 4408   Lim wlim 4409   suc csuc 4410   Compccmp 17129
This theorem is referenced by:  ordcmp  24958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-fin 6883  df-cmp 17130
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