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Theorem nlimsucg 4317
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 4159 . . . . . 6 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 4314 . . . . . 6 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 141 . . . . 5 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 4160 . . . . 5 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
53, 4jca 294 . . . 4 (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = suc 𝐴))
6 ordtr 4142 . . . . . . . 8 (Ord 𝐴 → Tr 𝐴)
7 unisucg 4178 . . . . . . . . 9 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
87biimpa 284 . . . . . . . 8 ((𝐴𝑉 ∧ Tr 𝐴) → suc 𝐴 = 𝐴)
96, 8sylan2 274 . . . . . . 7 ((𝐴𝑉 ∧ Ord 𝐴) → suc 𝐴 = 𝐴)
109eqeq2d 2067 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
11 ordirr 4294 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
12 eleq2 2117 . . . . . . . . . 10 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
1312notbid 602 . . . . . . . . 9 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
1411, 13syl5ibrcom 150 . . . . . . . 8 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15 sucidg 4180 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ suc 𝐴)
1615con3i 572 . . . . . . . 8 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1714, 16syl6 33 . . . . . . 7 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1817adantl 266 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1910, 18sylbid 143 . . . . 5 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
2019expimpd 349 . . . 4 (𝐴𝑉 → ((Ord 𝐴 ∧ suc 𝐴 = suc 𝐴) → ¬ 𝐴𝑉))
215, 20syl5 32 . . 3 (𝐴𝑉 → (Lim suc 𝐴 → ¬ 𝐴𝑉))
2221con2d 564 . 2 (𝐴𝑉 → (𝐴𝑉 → ¬ Lim suc 𝐴))
2322pm2.43i 47 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101   = wceq 1259  wcel 1409   cuni 3607  Tr wtr 3881  Ord word 4126  Lim wlim 4128  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-ilim 4133  df-suc 4135
This theorem is referenced by: (None)
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