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Theorem nlimsucg 4550
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 4380 . . . . . 6 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 4547 . . . . . 6 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 133 . . . . 5 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 4381 . . . . 5 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
53, 4jca 304 . . . 4 (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = suc 𝐴))
6 ordtr 4363 . . . . . . . 8 (Ord 𝐴 → Tr 𝐴)
7 unisucg 4399 . . . . . . . . 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 2182 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
11 ordirr 4526 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
12 eleq2 2234 . . . . . . . . . 10 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
1312notbid 662 . . . . . . . . 9 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
1411, 13syl5ibrcom 156 . . . . . . . 8 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15 sucidg 4401 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ suc 𝐴)
1615con3i 627 . . . . . . . 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 619 . 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 1348  wcel 2141   cuni 3796  Tr wtr 4087  Ord word 4347  Lim wlim 4349  suc csuc 4350
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-ilim 4354  df-suc 4356
This theorem is referenced by: (None)
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