Users' Mathboxes Mathbox for Chen-Pang He < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsucncmp Structured version   Visualization version   GIF version

Theorem limsucncmp 33680
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 ∈ Comp)

Proof of Theorem limsucncmp
StepHypRef Expression
1 suceq 6253 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On))
21eleq1d 2901 . . 3 (𝐴 = if(Lim 𝐴, 𝐴, On) → (suc 𝐴 ∈ Comp ↔ suc if(Lim 𝐴, 𝐴, On) ∈ Comp))
32notbid 319 . 2 (𝐴 = if(Lim 𝐴, 𝐴, On) → (¬ suc 𝐴 ∈ Comp ↔ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp))
4 limeq 6200 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On)))
5 limeq 6200 . . . 4 (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On)))
6 limon 7542 . . . 4 Lim On
74, 5, 6elimhyp 4532 . . 3 Lim if(Lim 𝐴, 𝐴, On)
87limsucncmpi 33679 . 2 ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp
93, 8dedth 4525 1 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1530  wcel 2107  ifcif 4469  Oncon0 6188  Lim wlim 6189  suc csuc 6190  Compccmp 21912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-1o 8096  df-er 8282  df-en 8502  df-fin 8505  df-cmp 21913
This theorem is referenced by:  ordcmp  33681
  Copyright terms: Public domain W3C validator