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Theorem limsucncmp 32775
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 5931 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On))
21eleq1d 2835 . . 3 (𝐴 = if(Lim 𝐴, 𝐴, On) → (suc 𝐴 ∈ Comp ↔ suc if(Lim 𝐴, 𝐴, On) ∈ Comp))
32notbid 307 . 2 (𝐴 = if(Lim 𝐴, 𝐴, On) → (¬ suc 𝐴 ∈ Comp ↔ ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp))
4 limeq 5876 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On)))
5 limeq 5876 . . . 4 (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On)))
6 limon 7181 . . . 4 Lim On
74, 5, 6elimhyp 4285 . . 3 Lim if(Lim 𝐴, 𝐴, On)
87limsucncmpi 32774 . 2 ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp
93, 8dedth 4278 1 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wcel 2145  ifcif 4225  Oncon0 5864  Lim wlim 5865  suc csuc 5866  Compccmp 21403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7211  df-1o 7711  df-er 7894  df-en 8108  df-fin 8111  df-cmp 21404
This theorem is referenced by:  ordcmp  32776
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