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Theorem nlimsucg 4565
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 4395 . . . . . 6 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 4562 . . . . . 6 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 134 . . . . 5 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 4396 . . . . 5 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
53, 4jca 306 . . . 4 (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = suc 𝐴))
6 ordtr 4378 . . . . . . . 8 (Ord 𝐴 → Tr 𝐴)
7 unisucg 4414 . . . . . . . . 9 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
87biimpa 296 . . . . . . . 8 ((𝐴𝑉 ∧ Tr 𝐴) → suc 𝐴 = 𝐴)
96, 8sylan2 286 . . . . . . 7 ((𝐴𝑉 ∧ Ord 𝐴) → suc 𝐴 = 𝐴)
109eqeq2d 2189 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
11 ordirr 4541 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
12 eleq2 2241 . . . . . . . . . 10 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
1312notbid 667 . . . . . . . . 9 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
1411, 13syl5ibrcom 157 . . . . . . . 8 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15 sucidg 4416 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ suc 𝐴)
1615con3i 632 . . . . . . . 8 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1714, 16syl6 33 . . . . . . 7 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1817adantl 277 . . . . . 6 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
1910, 18sylbid 150 . . . . 5 ((𝐴𝑉 ∧ Ord 𝐴) → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
2019expimpd 363 . . . 4 (𝐴𝑉 → ((Ord 𝐴 ∧ suc 𝐴 = suc 𝐴) → ¬ 𝐴𝑉))
215, 20syl5 32 . . 3 (𝐴𝑉 → (Lim suc 𝐴 → ¬ 𝐴𝑉))
2221con2d 624 . 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 104   = wceq 1353  wcel 2148   cuni 3809  Tr wtr 4101  Ord word 4362  Lim wlim 4364  suc csuc 4365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-iord 4366  df-ilim 4369  df-suc 4371
This theorem is referenced by: (None)
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