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Theorem limsucncmp 25903
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 4580 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
21eleq1d 2446 . . 3  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( suc 
A  e.  Comp  <->  suc  if ( Lim  A ,  A ,  On )  e.  Comp ) )
32notbid 286 . 2  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( -. 
suc  A  e.  Comp  <->  -.  suc  if ( Lim  A ,  A ,  On )  e.  Comp ) )
4 limeq 4527 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
5 limeq 4527 . . . 4  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
6 limon 4749 . . . 4  |-  Lim  On
74, 5, 6elimhyp 3723 . . 3  |-  Lim  if ( Lim  A ,  A ,  On )
87limsucncmpi 25902 . 2  |-  -.  suc  if ( Lim  A ,  A ,  On )  e.  Comp
93, 8dedth 3716 1  |-  ( Lim 
A  ->  -.  suc  A  e.  Comp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   ifcif 3675   Oncon0 4515   Lim wlim 4516   suc csuc 4517   Compccmp 17364
This theorem is referenced by:  ordcmp  25904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-1o 6653  df-er 6834  df-en 7039  df-fin 7042  df-cmp 17365
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