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Theorem limsucncmp 31615
 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 5707 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On))
21eleq1d 2672 . . 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 5652 . . . 4 (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On)))
5 limeq 5652 . . . 4 (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On)))
6 limon 6928 . . . 4 Lim On
74, 5, 6elimhyp 4096 . . 3 Lim if(Lim 𝐴, 𝐴, On)
87limsucncmpi 31614 . 2 ¬ suc if(Lim 𝐴, 𝐴, On) ∈ Comp
93, 8dedth 4089 1 (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  ifcif 4036  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845  df-cmp 21000 This theorem is referenced by:  ordcmp  31616
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