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Theorem nlimsucg 4525
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 4355 . . . . . 6 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 4522 . . . . . 6 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 133 . . . . 5 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 4356 . . . . 5 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
53, 4jca 304 . . . 4 (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = suc 𝐴))
6 ordtr 4338 . . . . . . . 8 (Ord 𝐴 → Tr 𝐴)
7 unisucg 4374 . . . . . . . . 9 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
87biimpa 294 . . . . . . . 8 ((𝐴𝑉 ∧ Tr 𝐴) → suc 𝐴 = 𝐴)
96, 8sylan2 284 . . . . . . 7 ((𝐴𝑉 ∧ Ord 𝐴) → suc 𝐴 = 𝐴)
109eqeq2d 2169 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
11 ordirr 4501 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
12 eleq2 2221 . . . . . . . . . 10 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
1312notbid 657 . . . . . . . . 9 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
1411, 13syl5ibrcom 156 . . . . . . . 8 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15 sucidg 4376 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ suc 𝐴)
1615con3i 622 . . . . . . . 8 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1714, 16syl6 33 . . . . . . 7 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1817adantl 275 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1910, 18sylbid 149 . . . . 5 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
2019expimpd 361 . . . 4 (𝐴𝑉 → ((Ord 𝐴 ∧ suc 𝐴 = suc 𝐴) → ¬ 𝐴𝑉))
215, 20syl5 32 . . 3 (𝐴𝑉 → (Lim suc 𝐴 → ¬ 𝐴𝑉))
2221con2d 614 . 2 (𝐴𝑉 → (𝐴𝑉 → ¬ Lim suc 𝐴))
2322pm2.43i 49 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wcel 2128   cuni 3772  Tr wtr 4062  Ord word 4322  Lim wlim 4324  suc csuc 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-ilim 4329  df-suc 4331
This theorem is referenced by: (None)
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