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Theorem nlimsucg 7189
 Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg (𝐴𝑉 → ¬ Lim suc 𝐴)

Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 5927 . . . 4 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 7161 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 224 . . 3 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 5928 . . 3 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
5 ordunisuc 7179 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
65eqeq2d 2781 . . . 4 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
7 ordirr 5884 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
8 eleq2 2839 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
98notbid 307 . . . . . 6 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
107, 9syl5ibrcom 237 . . . . 5 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11 sucidg 5946 . . . . . 6 (𝐴𝑉𝐴 ∈ suc 𝐴)
1211con3i 151 . . . . 5 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1310, 12syl6 35 . . . 4 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
146, 13sylbid 230 . . 3 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
153, 4, 14sylc 65 . 2 (Lim suc 𝐴 → ¬ 𝐴𝑉)
1615con2i 136 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1631   ∈ wcel 2145  ∪ cuni 4574  Ord word 5865  Lim wlim 5867  suc csuc 5868 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-pr 5034  ax-un 7096 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-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872 This theorem is referenced by:  tz7.44-2  7656  rankxpsuc  8909  dfrdg2  32037  dfrdg4  32395
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